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Let x equal one of the digits. And let y equal the other digit.
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You are told that the sum of the two digits is 12. So you can write an equation to express this as:
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x + y = 12
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Next you are told that the difference of the two digits is 4. In equation form this is:
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x - y = 4
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At this point you have a system of two independent equations that can be used to solve for the two unknowns.
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x + y = 12
x - y = 4
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If you add the two equations vertically, note that the +y and the -y will cancel each other and you are left with:
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2x = 16
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Solve for x by dividing both sides by 2 to get x = 8
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If one of the digits is 8 and the sum of the two digits is 12, then by subtraction you know that the other digit must be 4. So the two digits are 8 and 4.
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From your answers for the digits you know that the answer formed from the two digits of 8 and 4 must be either 48 or 84. But you were told that the two-digit number you are looking for is less than 50. So the answer to this problem must be 48.
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Hope this helps you.

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Let x be the original unknown number.
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Double that number means taking 2 times x which is written as 2x.
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Then add 12 to that and you have 2x + 12
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The problem tells you that this should equal 1/2 of the original number, or (1/2)x
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Setting these two quantities equal results in the equation:
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2x + 12 = (1/2)x
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You can get rid of the fraction on the right side by doubling both sides of this equation ... Do that by multiplying both sides (all terms) by 2 and the equation becomes:
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4x + 24 = x
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Subtract x from both sides:
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3x + 24 = 0
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Then subtract 24 from both sides:
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3x = -24
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Finally solve for x (the original number) by dividing both sides of this equation by 3 to get:
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x = -24/3 = -8
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So the answer is that the original unknown number is -8.
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Check it out. Double it to get -16 then add 12 to the -16 and the result is -4. This checks because it is supposed to equal half the original number which is one-half of -8. And it does.
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This checks the answer. The original number is -8.
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Hope this helps you to understand the problem.
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Given:
.

.
First, move the -512 to the other side of the equation by adding +512 to both sides as follows:
.

.
On the left side the -512 and the +512 total zero. Therefore, you are left with the following equation:
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Solve for by dividing both sides by 27 and you get:
.

.
Now note that 512 is equal to 8 cubed. Also note that 27 is equal to 3 cubed. You can verify these two statements by using a calculator to first multiply 8 times 8 times 8 to get 512. Then use it to multiply 3 times 3 times 3 and you get 27. So by substituting 8 cubed for 512 and 3 cubed for 27, you can write the equation as:
.

.
By the power rule of exponents you can furthermore write the right side of this equation as shown:
.

.
Now solve for x by taking the cube root of both sides to get:
.

.
which simplifies to:
.

.
You can check this answer by returning to the equation that you were originally given and substitute for x. You should then see that:
.

.
becomes (after cubing the ):
.

.
and by cancelling the 27 in the numerator with the 27 in the denominator, this equation reduces to:
.

.
Since the left side does equal the right side, the answer that is correct.
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Hope this helps you to see how to do this problem.

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Each consecutive odd integer is separated by adding 2 to its preceding odd integer. For example: odd integers 3, 5, 7, and 9 are consecutive odd integers. Note that 5 is 3 + 2 and 7 is 5 + 2 and 9 is 7 + 2.
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Therefore, if N is an odd integer, the next consecutive odd integer is N+2 and the next consecutive odd integer after that is (N+2) + 2 and this can be simplified to N + 2 + 2 which is N+4. So we can say that three consecutive odd numbers are N, N+2, and N+4.
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The problem then describes three terms as follows:
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The first term is the first odd integer or N.
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The second term is twice the second odd integer or twice N+2. If you multiply N+2 by 2 the result is 2N+4.
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The third term is 4 times the third odd integer. This would be 4 times N+4. If you multiply 4*(N+4) the result is 4N+ 16.
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You are then to add these three terms and you are told that the answer is 97. When written in equation form this is:
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N + (2N + 4) + (4N + 16) = 97
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Group the terms containing N and separately group the constants and the equation becomes:
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(N + 2N + 4N) + (4 + 16) = 97
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Add the terms containing N and also add the constants and the equation simplifies to:
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7N + 20 = 97
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Get rid of the 20 on the left side of the equation by subtracting 20 from both sides to get:
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7N + 20 - 20 = 97 - 20
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and the subtractions simplify this equation to:
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7N = 77
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Solve for N by dividing both sides by 7 and the result is:
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N = 11
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So now we know that N, the first odd integer, is 11. This means that the second consecutive odd integer is 2 greater than 11 or it is 13 and the third consecutive odd integer is 2 greater than 13 or it is 15.
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So the three consecutive odd integers you were to find are 11, 13, and 15.
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Check this by adding the first odd integer (11) to two times 13 or (26) to 4 times 15 or (60).
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11 + 26 + 60
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If you add these three numbers, the sum is 97 just as the problem said it should be. Therefore, the answer is correct. The three consecutive odd integers the problem was looking for are 11, 13, and 15.
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Hope this helps you to understand the problem.

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You are correct that the answer to the question about distance is 30 miles.
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In solving that problem, you can use a PROPORTION as follows:
.

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You can solve this by cross multiplying and setting the products equal.
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Cross multiplying means first to multiply the numerator on the left side times the denominator on the right side. So multiply 60 (the left side numerator) times 1/2 (the right side denominator) to get 30.
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The second step in cross multiplying is to multiply the denominator on the left side times the numerator on the right side. So multiply 1 (the left side denominator) times x (the right side numerator). The product is 1x or just x for short.
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Finally, set the two products equal and solve for the unknown. So in setting the two products equal you have:
.

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No other steps are needed because this is already the value for x.
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Notice the PROPORTION we set up has miles as the units for both the left and right side numerators and hours as the units for both the left and right side denominators. That's a check. The numerators on both sides should be in the same units and the denominators on both sides should also have units that agree as well. The location of the unknown can appear in any of the 4 locations - left numerator, left denominator, right numerator, or right denominator. It all depends on what you are trying to solve for. In this case we were trying to find the miles for 1/2 hour. So the x had to be in the numerator because we set up miles to be the two numerators. Then the x also had to be on the right side because it was related to the 1/2 hour which is the right side denominator.
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As far as the ten-letter word ... a clue is to look above for the word that is in all capital letters. (It appears twice.) I think that this is the 10-letter word you are looking for.
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Hope this helps you to understand the problem and how to solve problems that involve the 10-letter word.

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Let's try to use the hint so you can find out where you might be having difficulty. You are given:
.

.
Before we begin, take note of the fact that this equation has a negative sign on the left side, but the right side is positive. For this equation to be true, the answer to taking the square root of (5 - 2U) must be negative so the negative sign in front of the radical makes the entire left side positive like the right side is. More on this later.
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Let's square both sides:
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Note that on the left side I've included the minus sign that precedes the square root radical. It (the minus sign) gets squared also, so after squaring, it becomes a plus sign.
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I strongly suspect that this may be the source of your problem.
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Also recall that when you square a quantity that is under the square root radical, you just get the quantity itself. For example:
.
and
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So for the problem you were given when the entire left side is squared (both the negative sign and the radical) it becomes:
.

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and the when the right side is squared it becomes . So after both sides are squared the equation becomes:
.

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Subtract 5 from both sides:
.

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Which simplifies to:
.

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To solve for U divide both sides by -2 to get:
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And after dividing out the left side we get the answer:
.

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Let's check the answer by substituting it into the original equation:
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Start with:
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Substitute -218 for U to get:
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Do the multiplication of -(2)*(-218) = + 436 under the radical:
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Add the two terms under the radical on the left side and you get:
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When you take the square root of 441 you can consider two possible answers: plus 21 and minus 21. (Normal convention is that the square root radical will give just positive answers.) However, if you square either +21 or -21 the answer will be +441. Since we have the minus sign before the radical, in order to get the equation to be true, we must use the -21 answer and discard the +21 answer. With that understanding, the equation will balance using U = -218.
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The critical point to be made is that with U = -218 we can still get the equation to be true, but to do so we have to consider ignore the positive answer created by the square root radical and allow for negative answers to be considered. Barring that, this problem cannot be solved for U. The original source of the problem with the negative sign occurs because in solving for U we needed to square the minus sign when we squared the entire left hand side.
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The problem really gets "messy" if you don't square the minus sign (-1) when you square the entire left side of the original problem. If you square just the radical and not the minus sign in front of the radical, the left side squared becomes:
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which becomes
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When you set that equal to the right side squared the equation becomes:
.

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Add +5 to both sides and this simplifies to:
.

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and dividing both sides by 2 the answer becomes:
.

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In checking this answer by substituting +223 for U in the original equation the term under the radical becomes which simplifies to a negative number 5 - 446 = -441. Until you learn about complex numbers (real and imaginary parts), taking the square root of a negative number is not possible. Certainly it does not apply to this problem since the right side of the equation (21) is a real number. This difficulty with handling the negative sign may have been the source of your difficulty.
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Hope this helps you ...

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There are three different forms of graphs in this problem.
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The first form is represented by the equation . This is called the slope intercept form and the standard general form of this equation is written as . In this general form, m which is the multiplier of the x is the slope of the graphed line. If m is positive, the graphed line goes upward as you move toward the right on the graph. But if m is negative, the graphed line goes downward as you move toward the right on the graph. Regardless of the sign, the amount of the slope is determined by saying "as I move horizontally to the right the number of units in the denominator of m, the graph goes up or down (depending on the sign) the number of units in the numerator of m. (If m is a whole number, the denominator is 1 and the numerator is the whole number.) The second thing to notice is the term . The value of + b is the point on the y-axis where the graph crosses the axis.
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In the first problem, m (which is the multiplier of x) is . The minus sign tells us that the graph slants downward as you move toward the right. It also tells us that as you move horizontally 4 units from any point on the graph, at the place you reach the end of the 4 units you go down 3 units and at that location you have another point on the graph. Where is a point on this graph. Look at + (-1). The -1 is the b that is added to the x term in the equation you were given. b is the point on the graph where the graph crosses the y-axis. Therefore, this graph crosses the y-axis at -1 on the axis. From this point move 4 units (slope denominator) horizontally and then down 3 units vertically (slope numerator) and mark that location. This location is a second point on the graph. From that second point move horizontally 4 units and then vertically down 3 units and you have the third point on the graph. When you finish you should have a graph that looks like this:
.

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Notice that from the y-axis intercept at -1, if you move 4 units horizontally to the right you will be at the point (+4, -1). Then if you move vertically down 3 units you will be at the point (4, -4) and that is a second point on the graph. Then you can move 4 more units horizontally to the right from this second point and you will be at (8, -4). Then go vertically down 3 units and you will be at the point (8, -7) and that is a third point on the graph.
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That pretty much is the first problem. All the other problems involve X = a constant and y = another constant.
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x = a constant means that no matter what value y may take, x will always be the same. The graph of this will be a vertical line that crosses the x-axis at the constant. Similarly, for y = a constant, that means that no matter what value x is, the corresponding value of y is the constant. The graph of this will be a horizontal line that crosses the y-axis at the constant.
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Second problem: x = -2 and y = 1.5. The two graphs are as shown:
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Notice that the vertical red line is the graph of x = -2 and the horizontal green line is the graph of y = +1.5. The common solution for these two graphs is the point where the red graph and the green graph cross at the point (-2, 1.5).
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Third problem: x=1 y=-1.75. The two graphs are as shown:
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Notice that the red graph is the graph of x = +1 and the green graph is the graph of y = -1.75.
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The pair of graphs has as a common solution the point (1, -1.75)
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Fourth problem: x=2 y=-2.5. The two graphs are:
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Make sure that you understand why these graphs look as they do and what the common solution for these two graphs is.
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Finally, the Fifth problem: x=-3 y=1.25
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Study this graph also so that you thoroughly understand how to graph x = constant and y = constant.
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Hope this helps you with understanding these problems.

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The slope intercept form, as you can see, is:
.

.
m, the multiplier of the x is the slope of the line that the slope intercept equation establishes.
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In order for a line to be parallel to a given line, it must have the same slope as the given line. The line that you were given is:
.

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Since -3/4 is the multiplier of x, it is the slope of the graph for given line. Therefore, what you are being asked to do is to find the slope intercept line that has a slope of and passes through the point (5, -1). So let's begin with the general slope intercept form:
.

.
Just for your info, a slope of can be interpreted as follows: The minus sign tells you that as the line moves to the right the graphed line drops downward. Had the sign been + the graphed line would go upward as you moved to the right. The fraction can be determined as follows. The denominator (4) tells you that for every 4 units the graph moves to the right, the drop or rise is the numerator (in this problem it is a drop of 3 units). If the multiplier of x is a whole number, for example 5, think of it as a numerator of 5 and a denominator of 1 (that is so that for every 1 unit moved horizontally to the right, the change upward is 5 (if +) or downward 5 (if minus)).
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We've already determined that the graph of the line that we need has a slope of
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So substitute this into the slope intercept form and you have:
.

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What else do you know? You know that the given point (5, -1) must satisfy this equation if it is to be on the line. So we can substitute x = 5 and y = -1 into the equation to get:
.

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Now all you have to do is a little math and equation solving to find what b (the point where the graph crosses the y-axis) needs to be. Multiply out the first term on the right side and you have:
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Add to both sides and this reduces the equation to:
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Convert -1 to -4/4 so that you can combine the two left hand terms:
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.

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which simplifies to:
.

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Since you now have both m and b, you can write the equation of this new line as:
.

.

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and the graph of these two parallel lines looks like:
.

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The red line is the graph that you were given. (Note the negative slope (down and to the right) and that graph crosses the y-axis at y = +1 and indicated by the +1 value for b in the original equation.) The green line is the graph that we developed as given by the slope intercept form:
.

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Note here that the slope (multiplier of x) is again , but this time the value of b (the intercept on the y-axis) is .
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A great big thanks for taking the time to work with your daughter. More parents should be like you !!!
.
Hope this helps you in some small way.

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Let's see if we can't figure this problem out.
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Since it's not very well defined by the problem, I'm going to assume that the number of vehicles for the year 2000 represents the number of vehicles on December 31st of that year.
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First for the number of vehicles. Each year that goes by, the number of vehicles increases by 4.5% (that is by the decimal 0.045). So that on December 31st of the year 2001 the increase in the number of cars is 202M times 0.045. That means that at the end of the first year (the end of 2001) the number of cars is the 202M at the end of 2000 plus the increase of 202M times 0.045. In algebraic form this is:
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202M + (202M*0.045)
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Factor out the 202M and this expression for the number of cars at the end of the first year (that is at the end of 2001) becomes:
.
202M*(1 + 0.045) = 202M*1.045
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Next what happens during the second year? You start the second year with the number of vehicles at the end of the year 2001. We just determined it to be:
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202M*(1.045)
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So the increase by the end of the second year (the end of 2002) will be 0.045 times 202M*(1.045) and at the end of the second year the total number of cars will be what you had at the end of 2001 plus the increase of 0.045 times the number number of cars at the end of 2001. In algebraic form this is:
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202M*(1.045) + (202M*(1.045))*(0.045)
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Factor out (202M*1.045) and you have:
.
(202M*1.045)*(1 + 0.045) = (202M*1.045)*(1.045) = 202M*(1.045)^2
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If you try this analysis for another year or two, it will become apparent that for each passing year, the number of cars increases by a factor of 1.045. This means that the number of cars at the end of a given year (call it V(t)) can be determined from the equation:
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V(t) = 202M*(1.045)^(t)
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Where t is the number of years after the year 2000. So for example in the year 2003, t would be 2003 minus 2000 or t would equal 3.
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So on December 31 of the year 2005 the number of cars would be:
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V(t) = 202M*(1.045)^t = 202M*(1.045)^(2005 - 2000) = 202M*(1.045)^5
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You can use a calculator to determine that 1.045 raised to the exponent 5 is 1.246181938 and when you multiply this by 202M the answer becomes 251.728751M.
At the end of the year 2005 the number of cars will be 251,728,751
.
The same type of analysis can be done for the population. (Again assume that the population each year is for the last day of that year.) The difference is that each year the increase is 1% or 0.01. And the population in the year 2000 is 258M on December 31st 2000. So by introducing these changes into the equation for V(t) we can say that the equation for the population at the end of a given year is:
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P(t) = 258M*(1.01^t)
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where t is again defined as the year of interest minus 2000. If you want to find the population on December 31st of 2004, t would be 2004 - 2000 or t would be 4.
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Now to find the year when the number of vehicles equals the population so that there is one vehicle per person on average. This is done by setting the right side of the equation for V(t) equal to the right side of the equation for P(t) and then solving for t. In other words:
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202M*(1.045^t) = 258M*(1.01^t)
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Our goal is to get terms containing t on the left side of the equation, and all other terms on the right side. Begin by dividing both sides of this equation by 202M and you get:
.
1.045^t = (258M/202M)*(1.01^t)
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Divide the 258M by 202M and you get 1.277227723. Substitute this into the equation and it becomes:
.
1.045^t = (1.277227723)*(1.01^t)
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Divide both sides by 1.01^t and the equation becomes:
.
(1.045^t)/(1.01^t) = 1.277227723
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Note on the left side that the exponent in the numerator is the same as the exponent in the denominator. Therefore, by the rules of exponents we can say:
.
(1.045/1.01)^t = 1.277227723
.
to simplify this a little, divide 1.045 by 1.01 and you have 1.034653465. Substitute this and the equation becomes:
.
(1.034653465)^t = 1.277227723
.
Any time you have a variable in an exponent, you should consider taking the log of both sides so that the variable can be brought out as a multiplier of the log. Let's use log base 10 since we can readily use a scientific calculator to determine the logarithm. Take log base 10 of both sides:
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log((1.034653465)^t) = log(1.277227723)
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bring the t out as the multiplier of the log:
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t*log(1.034653465)= log(1.277227723)
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Use a calculator to find the two logs:
.
t*(0.014794916) = 0.106268336
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Solve for t by dividing both sides of the equation by 0.014794916:
.
t = 0.106268336/0.014794916 = 7.182760136
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Since t is the year of interest minus 2000, we know that t occurs exactly 7.182760136 years after 2000 or to the nearest tenth, 7.2 years after 2000 which would be 2 tenths of the way into the year 2008. (Since a tenth of a year is 1.2 months, 2 tenths of the year should be 2.4 months into 2008 which would be around the middle of March 2008).
.
Check my work to ensure that I didn't make some dumb error or a "fat finger" calculator mistake.
.
Hope this helps you to understand the problem.

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Call the original price P. Then the store subtracts 35% of P from the price. Recall that 35% of P is the same as 0.35 times P. So the new price is:
.
P - 0.35P
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and the problem says that this new price is $38.58.
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So in equation form this becomes:
.
P - 0.35P = 38.58
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Combine the two terms on the left side by performing the subtraction to get:
.
0.65P = 38.58
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Solve for P by dividing both sides by 0.65 to get:
.
P = 38.58/0.65 = 59.35384615
.
And rounding off the answer to the nearest cent tells us that the original price of the radio was:
.
P = $59.35
.
Hope this helps you with learning how prices are marked down.

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Given to solve:
.

.
Let's begin by squaring both sides. When you square the left side you multiply the entire left side by itself. So the left side is squared by multiplying:
.

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We can do this multiplication by multiplying the first term in the first set of parentheses times both terms in the second set of parentheses as follows:
.

.
Note that the first term on the right side, when squared is just . Therefore, this first multiplication results in:
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<--- remember this answer
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Not done yet with squaring the left side. Next we have to multiply the second term in the first set of parentheses by both terms in the second set of parentheses. This multiplication is:
.

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Add this result to the answer we reminded ourselves to remember above:
.

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Add the two constants (the +2 and the +1) and the answer simplifies to:
.

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This is the left side squared. Now we have to return to the original equation you were given and square the right side. In other words we square . And when we square the square root of a quantity the answer is the quantity itself. So:
.

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This is the right side squared. It is equal to what we got for the left side squared. So the "left side squared" equals "right side squared" equation becomes:
.

.
We can simplify this by subtracting x from both sides of this equation. Doing this causes the x to disappear on both the left and right sides so we are left with the equation:
.

.
Next we transfer the 3 to the right side by subtracting 3 from both sides. This subtraction causes the 3 to disappear from the left side and the -5 on the right side becomes -8. So we are left with:
.

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Then divide both sides by -2 and the equation reduces to:
.

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We have just a radical on the left side and we can get rid of it by squaring both the left and right sides. That squaring causes the equation to become:
.

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Finally solve for x by subtracting 2 from both sides and the answer is:
.

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You can check this answer by returning to the original equation given to you in the problem and substituting 14 for x. You will find that this substitution causes both the left and right sides of the original equation to be equal. This is as follows:
.

.

.

.

.

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The answer checks. x equals 14.
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Hope this helps you to understand working with radicals a little better.

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Yes it can be. Here's the graph:
.

.
This graph shows that the ordered pairs (-1,-3)and (-4,-6) are on the line and the slope is 1 because for every unit the graph moves horizontally to the right it also moves up one unit in the vertical direction.
.
Good job!!!

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This problem does not make sense at all.
.
Think about it. At present the father is 32 and his son is 5. That means there is 27 years difference in their ages.
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There will always be 27 years difference in their ages ... now and forever more.
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This problem implies that as time goes by the difference in their ages gets smaller. In this case it implies that as time passes the difference in their ages reduces until it is only 10 years. Stretching that a little more, one might as well write the problem as:
.
"a father is 32 years old and his son is 5. How many years will pass before the father is the same age as his son?"
.
Do you really think that can happen?

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The locus of all the points that are located 5 units away from the point (4, -3) is a circle with the center located at the point (4, -3) and having a radius of 5. The equation for such a circle is:
.

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The two points where this circle intersects with the x-axis will have "y" values of zero. (Any point on the x-axis has zero as its corresponding "y" value.) And these are the two points that we are looking for.
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That being the case, set y = 0 in the circle equation and the equation reduces to:
.

.
Square out each of the terms in this equation to get:
.

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Combine the two constants on the left side and this equation becomes:
.

.
Subtract 25 from both sides:
.

.
On the left side the +25 and -25 sum to zero, and the same thing happens on the right side. This reduces the equation to:
.

.
Factor out an x on the left side and the equation becomes:
.

.
Note that this equation will be true if either of the factors is equal to zero. This is because when one of the factors is equal to zero, the left side gets multiplied by zero, and this makes the left side equal to the zero on the right side.
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Therefore, the equation will be correct if either:
.

.
or
.

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and in this second equation when you add + 8 to both sides it becomes:
.

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This means that the points 0 and +8 on the x-axis are 5 units from the point (4, -3). In the form of ordered pairs the answers will be that the points (0, 0) and (8, 0) are the two points that are on the x-axis and are 5 units from the given ordered pair (4, -3).
.
Hope this helps you to understand this way of using the distance formula in the form of the equation for a circle to solve this problem.

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Sometimes the answer is a little easier to find if you put the equation into the slope intercept form. The slope intercept form is:
.

.
In this slope intercept form the value of m, that is the multiplier of x, is the slope of the graph line. And the value of +b is the value on the y-axis where the graph crosses.
.
The equation you were given is:
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Let's move the 4x to the other side of this equation by subtracting 4x from both sides of the equation as follows:
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On the left side the 4x and the -4x cancel each other out and we are left with the equation:
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Notice that if you compare this with the slope intercept form, it is in exactly the same format. For this equation m, the multiplier of the x, is -4. The minus sign tells you that the slope is negative, meaning that as you move to the right on the graph, the line goes downward. The downward slope is 4. That means that every unit you move to the right along the x-axis the corresponding change in the value of y is 4 units. Since the slope is downward for this problem, that means that every unit you move to the right along the x-axis the value of y goes down 4 units.
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One point on the graph is +b or the constant term in the slope intercept form. It is the point on the y-axis where the graph crosses. In your problem, the value of +b is +0. That means that the graph crosses the y-axis at y = 0 and this is the origin.
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So you can start with a point at the origin, and then as you move 1 unit to the right along the x-axis the graph drops 4 units in the value of y. So when x = -1, the value of y drops to -4. Then as you move another unit to the right along the x-axis (you are now at x = 2), the value of y drops another 4 units from y = -4 down to y = -8. This is the point x = 2 and y = -8.
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You now have the following three ordered pairs on the graph: (0, 0), (1, -4), and (2, -8). Plot these three points and draw an extended line through them to get the graph. When you do that it should look like this:
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Hope this helps you to understand graphing of linear functions a little better.
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I notice that your mathematical expression uses "x" as meaning multiplication.
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You can set this up as an equation. When you do it looks like this:
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28 - 3 x 3 + 4 = 23
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You can make this an equality (both sides are equal) by inserting one set of parentheses as follows:
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28 - (3 x 3) + 4 = 23
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Inside of the parentheses the 3 x 3 equals 9. Substituting 9 for the contents inside the parentheses changes the left side to:
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28 - 9 + 4 = 23
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If you algebraically total the three numbers on the left side by subtracting 9 from 28 and then adding 4 to that answer, the equation becomes:
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23 = 23
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So the answer to your problem is to insert parentheses as shown below and the result will equal 23:
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28 - (3 x 3) + 4
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Hope this helps you understand the problem better.

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You can get rid of the "a"s in the denominator by multiplying all terms on both sides by a as follows:
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The a in the numerator cancels with the a in the denominator as follows:
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and what's left is:
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Subtract 4a from both sides:
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and this simplifies to:
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Next subtract 1 from both sides:
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On the left side the +1 and -1 cancel each other out and this reduces the equation to:
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Solve for a by dividing both sides by -2:
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and the answer becomes:
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That's the answer to this problem.
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Hope this helps you to see your way through the problem.

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The integers are consecutive. Therefore, each integer is obtained by adding 1 to the immediately preceding integer.
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So if n+0 is the first integer, then n+1 is the second integer and the third integer is n+1+1 or n+2, and so on. The first six of the integers are:
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n + 0
n + 1
n + 2
n + 3
n + 4
n + 5
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The sum of these 6 numbers is 6n + (0+1+2+3+4+5) which simplifies to 6n + 15.
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The problem states that the sum of the first 6 numbers is 381. So we can set up the equation:
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6n + 15 = 381
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Get rid of the 15 on the left side by subtracting 15 from both sides as follows:
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6n + 15 - 15 = 381 - 15
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On the left side the + 15 and the -15 cancel each other out and on the right side 381 - 15 = 366. So the equation simplifies to:
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6n = 366
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solve for n by dividing both sides by 6 to get:
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n = 366/6 = 61
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We now know that the first number is 61. Therefore, because they are consecutive, the first 6 numbers are 61, 62, 63, 64, 65, and 66. That means that the seventh through twelfth numbers are 67, 68, 69, 70, 71, and 72. Adding these 6 numbers:
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67 + 68 + 69 + 70 + 71 + 72
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results in the total of 417 and that's the answer to this problem.
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Hope that this helps you to understand the problem.

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You have a minor difference because of rounding differences. Rather than use a table, you can use the following equation:
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Where the letters represent the following:
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P is the future value of an investment
C is the amount of money that you start with in the investment
R is the annual rate of interest in decimal form (example: 6% is entered as 0.06)
N is the number of times per year that the interest will be compounded
T equals the number of years into the future that the investment will be determined
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For this problem you are going to determine P. The money to be invested C is $4,000. The interest rate R is 0.04 annually. The interest will be applied only once per year at the end of the year so N = 1. And finally, the investment period T is 5 years. Substituting these values into the equation gives:
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Simplifying the terms in parentheses results in:
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and adding the 1 + 0.04 in the parentheses gives:
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Using a calculator to determine 1.04 raised to the 5th power simplifies the equation to:
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and multiplying out the right side of this equation gives you the answer:
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When you round to the nearest cent the answer becomes $4866.61
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That's all there is to this problem. But in the way of further explanation, a Certificate of Deposit (or CD for short) is a contract that if you deposit money for a specified period of time it will pay you a guaranteed annual rate of interest for the entire period. (It will also tell you how many times the year the annual rate of interest is to be compounded during each year.) Finally it specifies how long in terms of years or fractions of years the money plus any interest that is paid is to remain invested. At the end of the time that the money is to be invested the Certificate of Deposit expires (or is said to reach maturity) and at that time the investor will be paid the original amount invested plus any interest that it has earned. For this problem the CD matured at the end of the fifth year.
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Hope this helps you to build your confidence about what you are doing. You understood the meaning of the problem and you went about solving it correctly. Congratulations and keep up the good work!!!!

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This is a very interesting situation. It sounds as if this problem were designed by either a mathematician or an educator who never has had one day of practical experience as a salesman, especially as a flooring salesman. Let's discuss a little bit each of the six questions (A through F) that you have been presented with.
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Question A. What is the main question(s) that you think the floor salesman is being asked by Mrs. Cooper?
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Unless Mrs. Cooper is a woman to whom a dollar (or hundreds or thousands of dollars) is of little concern, I can guarantee that the primary question she has starts with, "How much will it cost me to ...?" If she is astute, she also will ask questions about durability of marble tile, what work or cost is required to maintain it, how long can be expected to last, and how much of the cost of this job could she expect to recapture if she later decided to sell the house? She may also want to know if she can save money by switching from 9" square tiles to 12" squares or 18" squares or some other dimensions. She may want to know if the green color is difficult to find. If she is an environmentalist, she may ask where the marble is being quarried in search of the nearest source that does not involve the environmental impact of long distance transportation. For example, does the tile that she is specifying need to be imported from a great distance away. She may ask to see actual samples of the marble. She may ask if the tile is gauged and calibrated (of uniform thickness at the edges and flat across the back side as well as on the upward face). She may ask if the salesman would recommend a highly polished exposed side. In addition, Mrs. Cooper may want to know how long it will take to gather all the supplies and to schedule the work as well as how long the actual work in her house can be expected to last. She might ask how much dirt or dust can she expect from the installation process and how the installers will help to minimize that nuisance. (Wait until she sees the dust involved with wet sawing or grinding marble tile.) Many other questions are likely to arise as she talks to the salesman.
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Question B. Identify at least two facts in the problem statement that are not relevant to finding the solution to this problem.
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To the salesman, every statement in the problem is relevant to finding a solution. For example, even the fact that the customer's name is Mrs. Cooper is relevant. If he called her Mrs. Smith or Miss Jones or "honey" or something else inappropriate, he is well on his way to losing a sale. (If she's done her homework, she may even know more about the job than the salesman does. Just ask a Home Depot or Lowe's clerk detailed questions about floor tile.) The fact that the room is a den is important because it will give the salesman a clue of what type of traffic the floor is likely to experience. The color green is important because it may well determine how available such tiles are and where the quarry is. The fact that Mrs. Cooper has asked for 9" square tiles is germane since that is not a common size. Therefore, this size may require a special order which will add to cost, time, and availability. (Common sizes are 3" x 6", 4" x 8", 8" x 8", 12" x 12", 18" x 18", 16" x 24".) The fact that marble is replacing carpet is also important. If the marble tile (usually 3/8" but may be 1/2" thick) is not as thick as the carpet and pad that currently are on the floor, the removal of the carpet can create a height difference with the baseboard around the room and with the existing door jamb casement as well.
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The salesman is in a battle with his competition to prove to the customer that he is the one she should trust because he is thorough, knowledgeable, a person who caters to her need for flooring and what she wants, and is presenting a quality solution that is cost competitive. He also needs to give her confidence that the product he is selling to her will be a credit to the appearance of her house and to her taste in decorating. He also needs to know and understand the impact of everything involved so that he can sure all the costs are appropriately identified and managed.
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Question C. Name at least three contextual or mathematical clues that the floor salesman will have to use to interpret this problem situation.
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Since it is very basic to the fundamental cost of the project, the salesman needs to determine if Mrs. Cooper really means marble tile or whether she might be using marble as a generic term for any natural stone tile (slate, granite, marble, sandstone, travertine, limestone, etc.) or maybe even for ceramic, glazed clay, glass, or other man-made tiles. The exact type of tile will impact the materials cost. Really good marble tile is expensive. Some is in the vicinity of $20 US per square foot and for this job (requiring approximately 200 square feet of tile as discussed below) the cost of the tile will be $4000 US before adding costs for all the other materials, as well as labor, and miscellaneous expenses.
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Obviously the square footage of the floor (12' x 15' = 180 sq ft) is important as a clue. So is the fact that Mrs. Cooper has asked for 9" square tiles since the size of the tiles will help determine how many are needed to cover the floor. (However, marble floor tile is generally priced in terms of square feet, not in terms of tile count.) Not implicitly stated in the problem is the width of the grout spaces that Mrs. Cooper would like between tiles. This has an impact because the wider the grout lines, the more the grout mix and labor will be affected. The narrower the grout lines, the more floor space that the marble tile itself will have to cover with an accompanying reduction in the amount of grout mix required. Also, not covered will be the excess tile that will be required in case a tile breaks or is cut wrong. Although the tile is to cover the 180 square feet of the den, the salesman (if he is smart) will probably add about 10% overage to cover such unforeseen contingencies, to allow for throwaway material along room edges where only half tiles may be needed, and to give spares to the customer in case in the future a tile cracks or is stained and needs to be replaced. Replacement tiles should be from the same quarry and lot so that they match the damaged tiles. So although the salesman is to cover 180 square feet with tile, his likely order will be for approximately 200 square feet of tile.
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The salesman also needs to recognize the problems associated with installing tile. A carpet and its pad can cover and disguise a multitude of sins in the sub floor. Marble tile is not so forgiving. Because it usually has a polished surface, if all the tiles are not cemented down nearly uniformly flat, the ambient light will reflect differently from each tile and will make the floor look like it was haphazardly installed. This may make it necessary to level the tile sub flooring, and that would have to be factored into the mathematical calculations for cost and materials. The salesman has to understand the labor requirements such as number of team members, the hourly cost rate (including insurance and benefits) for each type of laborer involved, the expected duration of the job, material costs (for example tile, tile adhesive, sub floor treatments, grout, tool wear and tear, etc.), and hidden costs such as transportation to/from the job site.
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Question D. To clarify his thinking, what questions will the salesman have to consider before answering the question being posed?
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The previous discussion above has already identified some of the factors/questions that the salesman will have to consider before answering Mrs. Cooper's question about what will it cost. In addition the salesman will also need to determine the scope of the work to be performed. For example, will the installers be responsible for removing the existing carpet and pad and for disposing of it? Will the installation include any modifications to the baseboard situation to disguise any gaps that may be encountered as a result of the difference in the height of the carpet and pad as compared to the height of tile? Will the old baseboard have to be removed and replaced so that it matches the new height of the tile flooring? Will transitions have to be installed so the new flooring in the den will mate well with the existing flooring just outside the doors leading into the den?
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Question E. What additional information will the floor salesman need to resolve the problem situation?
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A lot of the discussion above relates to what the salesman needs to evaluate and resolve in order to price the job so that he stays in contention with the competition. On the other hand, he must price the job so it covers costs and contingency costs plus some profit for the company he represents.
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Question F. What mathematical models or formulas will the floor salesman have to use to determine the solution to this problem?
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Obviously, the mathematical model must consider the area of floor to be installed in the room. We've also recognized that this model will also likely include the cost of a 10% overage in the amount of tile to provide for contingencies and spares for potential future damage. The width of the grout between tiles is a factor that can be determined and priced.
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However, salesmen often have other models that affect the cost being proposed to the customer. For example, the salesman may say that in his experience, the average cost of materials represents 1/3 of the total cost and the cost of labor and other miscellaneous associated costs represents 2/3 of the total cost. Therefore, if he carefully estimates and computes the cost of the materials (call that cost M), he can say that 1/3 of the total cost of the job (call it C) can be computed from the equation:
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Multiplying both sides of this equation by 3 results in:
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So if he closely prices the material costs (M) he can get the cost to perform the entire job (C) by multiplying the material costs by 3. Then he can compute the labor and other miscellaneous costs as being be two-thirds of C.
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Other costs to the customer must be added to that amount. For example, if he wants the job to have a 10% profitability he needs to add 10% of C to the total cost to the customer. He also needs to add to the customer's bill any taxes that must be paid. For example, sales taxes that must be paid on the materials.
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Hopefully, this rambling, unproofread discussion will be of some help to you and will give you ideas of how to answer the questions posed by this problem. Feel free to modify the organization of your answers.
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Obviously real life is often considerably different from and more complex than what the educators and mathematicians think while designing problems meant to teach lessons. Might as well get used to it because that's the way that life is, and that's what makes life so really interesting as you travel through it.

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I think what you want to know is if you are given the equation:
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how can you graph it and particularly what does the form y = mx + b have to do with graphing this equation.
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The form:
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is a useful form because it can help you to understand what the graph of an equation looks like. (It is called the slope-intercept form.) The value of the constant b tells you the value on the y-axis where the graph crosses (or intercepts)the y-axis. The value m (which is the multiplier of x) tells you the slope (or slant) of the line. The slope tells you for every 1 unit you move to the right in the horizontal x direction, how many units does the straight line graph move up or down in the y direction. If m has a positive value the graphed line moves upward as you move to the right, and if m has a negative value the graphed line moves downward as you move to the right.
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With this information, we can learn something about the graph of the equation that you were given. The equation is:
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But it is not in the form of . However, if we could get it into the form then we would know some information about the graph.
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To get it into the form, we need to rearrange the given equation so that it has only y on the left side and so the x term is on the right side along with the constant. THIS IS THE IMPORTANT THING TO UNDERSTAND. If we rearrange the given equation until it is in the form then we gain some valuable information about the graph.
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To rearrange the equation into the slope-intercept form we can start by moving the term containing the y to the left side of the equation. We do this by subtracting 2y from both sides of the given equation as follows:
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On the right side the 2y and the minus 2y cancel each other and we are left with:
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Next we move the x term to the right side by subtracting x from both sides as follows:
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On the left side the x and the minus x cancel each other out and we are left with:
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We're getting closer to having things in the slope-intercept form. Notice that in the slope-intercept form we need to have +y (written as just y) on the left side. But in our form we have -2y. So let's get just y on the left side by dividing the entire equation, both sides and every term, by -2 as follows:
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The division by -2 simplifies as follows:
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and results in:
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This is the rearranged form of the original equation that we are looking for because it can be compared directly term by term with the slope-intercept form of .
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Note that on the right side +b which is the constant term compares to +(-2)in our rearranged form. Since b is the y-axis intercept, we can now tell by inspection that the point -2 on the y-axis is where the graph crosses.
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Next, notice that (1/2) is the multiplier of x. Since the multiplier of x in the slope-intercept form tells you the slope, we can tell that the slope of the graph we have is 1/2. It is positive which means that it slants upward as we move toward the right on the graph. It also tells us that for every unit we move horizontally to the right, the graphed line will simultaneously move vertically up 1/2 unit.
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And our graph will look as follows:
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In particular, notice that the graph crosses the y-axis at -2. Also notice that as you move to the right along the x-axis, the graph rises 1/2 unit for each unit along the x-axis. So when you move from the origin 4 units to the right on the x-axis, the graph rises 4 times 1/2 or 2 units upward in the y-direction.
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Hope this helps you to understand the slope-intercept form and how it can assist you in graphing equations.

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You are asked to solve for the values that h can have to satisfy the inequality:
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Your book or your teacher may have a different way of doing absolute value problems such as this one, but I'll show you a way that I like to do these.
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You will solve two separate inequalities. One inequality will be done by first eliminating the absolute value bars and putting a plus sign + in front of the whole quantity that was inside these absolute value bars. The second inequality will also be done by first eliminating the absolute value bars, but then you put a minus sign - in front of the whole quantity that was inside these absolute value bars.
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One other thing to remember. If you multiply or divide both sides of an inequality by a negative quantity, you need to reverse the direction of the inequality.
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This may sound a little complex, but it's really pretty easy to do.
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Start with the inequality you were given to solve for h:
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Eliminate the absolute value bars and assume that h has a plus sign in front of it:
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That's the first answer. h can be any value greater than +6 and that means that h can lie to the right of +6 on the number line.
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Now let's solve the second inequality. Again start with the inequality you were given to solve for h:
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Eliminate the absolute value bars and this time assume that h has a minus sign in front of it:
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But we need to solve for +h. In order to do that we can multiply both sides of the inequality by -1. Remember the rule, however. When you multiply both sides by a negative 1, you also have to reverse the direction of the inequality sign.
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The multiplication results in a plus h on the left side and a minus 6 on the right side as follows:
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and because of the multiplication of both sides by a negative number, we reverse the direction of the inequality sign and get:
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This means that h can also be any number less than -6 (or any number that is to the left of -6 on the number line).
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In summary, the answer to this inequality problem is that h must either be greater than +6 or it must be less than -6.
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To help you a little further with using this method, let's work another quick example. Suppose the problem you are given is:
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Very quickly, first eliminate the absolute value bars and put a plus in front of the quantity inside the bars to get:
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Since the parentheses are preceded by a presumed + sign, the parentheses can be removed without changing the signs of he terms within. This simplifies the inequality to:
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Subtract 3 from both sides. (NOTE: This negative quantity does not change the direction of the inequality sign because it is subtraction NOT multiplication or division.)
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and the subtraction on both sides results in:
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So one of the answers is that x must be less than 7 (or to the left of +7 on the number line).
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Now for the second solution. Return to the original problem:
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Eliminate the absolute value bars and put a negative sign in front of the entire quantity that was within the bars to get:
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Remove the parentheses by changing the signs on each of the terms that they contain. This results in:
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Add 3 to both sides:
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Multiply both sides by -1 to solve for positive x and don't forget to reverse the direction of the inequality sign. When you do those steps the resulting answer is:
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This means that x must be greater than -13 (or x is to the right of -13 on the number line).
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Combining the two answers we have x must be smaller than +7, but it must be larger than -13. So the values that x can have are between -13 and +7.
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I hope this example adds a little more explanation and provides you with a little more understanding on how to do absolute value problems. Please double check my work to ensure that I didn't make any dumb mistakes. It's too late and I probably should not be working math problems.

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OP is the entire line and the point M lies somewhere along the line between the endpoint O and the endpoint P.
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The sum of the lengths OM and MP equals the entire line OP. You are told that the length OM is:
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and the length MP is:
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Therefore, the some of these two line segments is the sum of the right sides of these two equations. In other words, the equation form of the sum of these two segments is:
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On the right side of this equation, the parentheses can be removed without changing the terms they contain because both sets of parentheses are preceded by a plus sign. This makes the equation become:
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Combine the two constants on the right side and the two terms containing x as follows:
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You are also told that the length of the line OP (as represented by OM + OP) is 44. From that you can write that the right side of this equation equals 44 as follows:
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Get rid of the +2 on the left side by subtracting 2 from both sides:
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and this simplifies to:
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Solve for x by dividing both sides by 3 to get:
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Now that we know the value of x is 14, you can find the length of OM (given by x+8) and the length of MP (given by 2x-6). Substitute 14 for x in both these and you get:
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which simplifies to
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and
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which first simplifies to and then to
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Note that the two line segments OM and MP have lengths of 22. Since they are equal in length, by the definition of midpoint as the point that divides a line into two equal segments, you can say that the point M is the midpoint of the line OP.
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Hope this helps you to understand the problem and the way that you can solve it.

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You were given to simplify using the product rule the multiplication of:
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The first thing you may want to do is to multiply the coefficients (the 7 times the 5) and you get as the product 35. This makes the problem become:
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Next you may multiply the x terms together. Since both of the "x" terms have exponents, you multiply them by adding their exponents making the result as follows:
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After you add the 6 + 2 the problem simplifies to:
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Finally, you can multiply the two "y" terms by adding their exponents. Remember that the term is the same as . Therefore in adding the exponents of the "y" terms you get:
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Adding the "y" exponents together results in the answer of:
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Since both the "x" and the "y" terms have the same exponent you can also express this answer as:
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This last form of the answer is an example of the power rule which basically says that if you have the product of factors inside parentheses and raise this product to a common power you can do so by raising each of the factors to the common power. (Only this answer is in reverse. We had factors raised to a common power of 8 and changed form by raising the product of the factors to the common power.)
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Hope this helps you to understand your problem and the product rule a little better.

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Because you have two unknowns and only one equation, you cannot in general get a numerical solution for x and for y. (This is an important understanding.) The best you can do is to solve for one of the variables in terms of the other variable. In this problem you are asked to solve for y.
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As in most algebraic solutions, a starting strategy is to isolate on one side of the equal sign the unknown variable that you are to solve for. So for this problem, since you are asked to solve for "y", get all the terms that contain "y" on one side of the equation and all the other terms on the other.
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The term 4x does not contain a "y". Move it to the other side of the equal sign by subtracting 4x from both sides of the equation as follows:
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and you are left with:
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The two terms on the left side both contain y as a multiplier. Therefore, you can factor a "y" from each of these terms to get:
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Now you can solve for "y" by dividing both sides by the multiplier of "y"s. That multiplier is the quantity (7 + 5x). Dividing both sides by this quantity results in:
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which reduces to simply:
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That's the answer. But a word of caution for this answer. Division by zero is not allowed in algebra. Therefore, the quantity (7 + 5x) cannot equal zero. In equation form this means:
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is not permitted. Solve this for x by subtracting 7 from both sides:
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and divide by both sides by the multiplier of x:
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This tells you that "x" is not allowed to equal because if it does, you will have a division by zero. Other than that, this problem tells you that the solution for "y" depends on the value of "x". You can select a value for "x" and then compute the corresponding value of "y". As one example of this:
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Suppose you select x = 1. Then the corresponding value of "y" can be determined as follows:
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Substitute 1 for x and you have:
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which reduces to:
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and this further simplifies to:
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Then by dividing both the numerator and the denominator by 4 the fraction becomes:
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This means that the coordinate or (x,y) point of (1,-1/3) is a solution of the problem and lies on the graph of the solution set for the equation that you were originally given in the problem.
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Hope this helps you to see the general process for solving this problem as well as some other considerations in the answer.

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First, you need to be careful that you don't average the two averages. By this I mean that you don't add the average score of the juniors (80) with the average score of the seniors (70) to get:
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This would give you an "overall average" of 75 and it would be wrong.
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Next, you can get a sense of the answer by recognizing that more students scored near 80 points than the number of students that scored near 70 points. Therefore, the "overall average" will be closer to 80 than it is to 70. So you can suspect that the answer of 73 (answer A) is also wrong.
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So how do you solve this problem?
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To have the 35 juniors get an average score of 80, the total of all the scores by juniors divided by the number of juniors (35) must be 80. So you can ask yourself what would be the total of the junior's scores (call it TJ) that when divided by 35 gives an answer of 80? In equation form this would be:
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This can be solved by multiplying both sides of this equation by 35 as follows:
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Multiply out the right side and you have:
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So the juniors scored a total of 2800 points.
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How many total points did the seniors score? (Call this total TS.) Using the same process for the 15 seniors as we just did for the juniors we would get:
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Multiply both sides of this equation by 15 as follows:
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Multiply out the right side and you have:
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So the seniors scored a total of 1050 points.
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Now when you combine the points scored by the juniors and seniors, the total number of points scored by the two classes was:
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So the 50 students scored a total of 3850 points. This means that the average score for the 50 students is the total points scored by all 50 students divided by the total number of students who took the test. In equation form this can be written as:
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The correct answer to this problem is answer D.
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I hope that this helps you to understand the problem and also helps you get a sense of how to work with averages. Good luck!
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Interesting problem. Try these:
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A little complex, but here goes with a method of arriving at a solution. This first part is formal way of arriving at the answer. After we get through this, I'll discuss a little easier way (at least I think it is easier to understand) that uses more number sense than relying on a formal equation.
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First, let's look at the equation for the sum of a geometric progression to n terms. The general form of the equation is (according to a math textbook):
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a + ar^1 + ar^2 + ar^3 + ... ar^(n-1) = (a[r^n - 1])/(r -1)
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Note that the right side of this equation gives the sum of this series.
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If you look at the problem you were given you will note that it progresses as follows:
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1 person tells 4. Then those 4 each tell 4 more and this adds 16 more people who know. Then those 16 each tell 4 more for a increase of 16 times 4 = 64. Then those 64 each tell 4 more for an increase of 256 who know.
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When you compare this to a geometric progression you can see that it is of that form ... namely let a = 1 and r = 4 and you have:
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1 + 1*4^1 + 1*4^2 + 1*4^3 + ... 1*4^(n-1)
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Therefore, the sum will be (a[r^n - 1])/(r -1) if a = 1, r = 4, and n is the unknown number of terms you are trying to determine. Substitute a = 1 and r = 4 into this sum and you have:
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(1[4^n - 1]/(4-1))
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The denominator (4 -1) is 3 and the 1 times a quantity is just the quantity. So the sum formula reduces to:
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[4^n - 1]/3
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But the problem wants the sum to be 5000, the total number of people who know the story. So set the sum equal to 5000 as follows:
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[4^n - 1]/3 = 5000
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Get rid of the denominator of 3 by multiplying both sides by 3 and the equation becomes:
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[4^n - 1] = 15000
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The brackets can be removed and then get rid of the -1 by adding 1 to both sides of the equation and you have:
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4^n = 15001
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We need to solve for n, an exponent. Let's take the logarithm of both sides. Use base 10 logs:
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log(4^n) = log(15001)
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The rules of logs tell you that you can bring the exponent out as a multiplier of the log. So bring the n out as a multiplier and you have:
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n*log(4) = log(15001)
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Use a scientific calculator to find that log(4) is 0.602059991 and log(15001) is 4.176120211. Substitute these values into the equation and you have:
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n*0.602059991 = 4.176120211
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Solve for n by dividing the right side by the multiplier of n which is 0.602059991:
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n = 4.176120211/0.602059991
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Do the division and you get the answer than n = 6.936385528
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This value of n would give you an answer of 5000 exactly. However, it would involve telling fractions of persons instead of whole people. So let's just say that n = 7 and we'll have a little more than 5000 people who know the story. As a matter of fact the sum will be:
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(a[r^n - 1])/(r -1) in which a = 1, r = 4, and n = 7.
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Substitute the values for a, r, and n and the sum becomes:
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(1*(4^7 - 1))/(4-1) = (4^7 - 1)/3 = (16384 - 1)/3 =16383/3 = 5461 people
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Now return to the left side of the equation for a geometric series. With a = 1 and r = 4 the left side becomes:
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1 + 1*4^1 + 1*4^2 + 1*4^3 + ... 1*4^(n-1)
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Remember that we said n was 7. This being the case our last term in the progression should have the exponent (n-1) which would be 6. Notice also that the a = 1 multiplier can be dropped because multiplying each term by 1 doesn't change anything. So the progression applicable to this problem would be:
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1 + 4^1 + 4^2 + 4^3 + 4^4 + 4^5 + 4^6
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Think about it. It took 20 minutes to get from the first term to the second term. Then it took another 20 minutes to get from the second term to the third term. An additional 20 minutes was needed to get from the third term to the fourth term. An so on. To get all the way to the last term in this progression from the very first term will require 6 transitions of 20 minutes each, for a total of 120 minutes. That means beginning with a single person knowing the story, 120 minutes later (2 hours) a total of 5461 persons know the story.
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Sort of complex, but that's a formal way of getting the answer.
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In a less formal way, you could have just written down the terms:
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1 + 4 + 16 + 64 + 256 + 1024 + 4096 + 16384 + ...
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which recognizes the fact that each 20 minutes increases the number of people who know the story by a factor of 4. To get from the first term to the second term, multiply the first term by 4. To get from the second term (4) to the third term, multiply the second term (4) by 4. To get from the third term (16) to the fourth term multiply the third term (16) by 4. To get from the fourth term (64) to the 5th term, multiply the fourth term (64) by 4. And so on.
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Since the last calculation is 16384 you have obviously exceeded 5000 by quite a bit. So stop and now use a calculator to begin adding terms. Adding all the numbers up from 1 through 4096 will give you the total of 5461 persons. It's a method that is probably faster and easier to understand, but a little less elegant than using the equation for the sum of a geometric progression of n terms. You still get the answer of 120 minutes (or 2 hours) to get from the 1 person knowing the story to the 5461 people knowing it. (There are 6 of the 20 minute jumps to get from the first term (1) to the last term (4096).
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Hope that I didn't confuse you with the math. Maybe you gained enough insight into the problem to see another logical way of getting the answer. In any case, check my work. It's very late and I'm out of coffee. So I may have made some dumb mistakes. Hope not. Good luck.

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Again, another dice problem. As I indicated in the answer to one of your other problems, there are 36 possible outcomes on the roll of a pair of fair dice. These outcomes are:
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1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
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Out of the 36 possible outcomes, how many of the dice rolls total 6 or 9. Count them up. You should find there are 9 of these "winners" beginning with 1,5 and ending with 6,3. So the probability of winning on the first roll by rolling a 6 or a 9 is the number of winners (9 of them) divided by the total number of possible outcomes (36). 9 divided by 36 reduces to 1 divided by 4 which has a decimal equivalent of 0.25 (or 25%). You have a one in four chance of winning on the first roll. This suggests that in the long run on every 4 rolls you are likely to be a winner for one of those rolls.
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Good luck. Just keep working at it and eventually it will start to make sense.
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If you roll a pair of fair dice, there are 36 possible outcomes. For each number you roll on the first die you have six possible numbers turning up on the second die. Six possible outcomes on the first die times 6 possible outcomes on the second die. Here are the 36 possible outcomes in the format of first die outcome, second die outcome:
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1,1
1,2
1,3
1,4
1,5
1,6 *
2,1
2,2
2,3
2,4
2,5 *
2,6 *
3,1
3,2
3,3
3,4 *
3,5 *
3,6 *
4,1
4,2
4,3 *
4,4 *
4,5 *
4,6 *
5,1
5,2 *
5,3 *
5,4 *
5,5 *
5,6 *
6,1 *
6,2 *
6,3 *
6,4 *
6,5 *
6,6 *
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I put an asterisk next to each outcome for which the total on the two dice is 7 or more than 7. Notice that if you count all the asterisks, there are 21 possible "winners" being that their total is 7 or more than 7 as required by the problem.
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The total number of possible outcomes is 36 and there are 21 possible "winners." Therefore, the probability of winning is 21 out of 36 or 21 divided by 36. If you do the division, the answer is 0.58333333 or you have a 58.333333% chance that on a single roll you will get a total of 7 or greater.
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Hope this helps you a little further along on your journey through probability.
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This is a problem very similar to those that involve rolling a die or drawing a specific card from a deck of cards. The method to use is look at the total number of possible outcomes and divide that number into the number of possible selections that would win.
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For example, a die has 6 possible outcomes. When you roll it you can get 1, 2, 3, 4, 5, or 6. What is the possibility that you will roll an even number? There are three winning numbers ... 2, 4, or 6 are the even numbers. Therefore, when you roll a die you have three possibilities to "win" out of six possible outcomes. The probability that you will win is 3 divided by 6 which is 1/2 or in decimal form is 0.50 (50 percent).
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Similarly, the odds of rolling an odd number (1, 3, or 5) are 3 chances out of 6 possible outcomes. So the odds of rolling an odd number are 3 divided by 6 or 0.50 also.
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Note that the odds of rolling either an even number or an odd number are 6 out of 6 or 1.00 meaning you have a 100% chance of winning. This is the same as the probability of rolling an even number (0.50) plus the probability of rolling an odd number (0.50) and that sum is 1.00.
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With cards. Given a fair deck (unmarked backs) of 52 cards, what is the probability of drawing a spade on a single draw? There are 13 spades in the deck, so on any single draw your chances of drawing a spade are 13 out of 52 possible cards. So the probability of drawing a spade is 13 divided by 52 which reduces to 1/4 and the decimal equivalent is 0.25. You have a 25% chance of drawing a spade on a single draw.
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Or how about on a single draw from the deck of 52, what are the odds that you will draw an ace? There are four aces in the deck, so the number of "winners" is 4 out of the 52 possible outcomes. Divide 4 by 52 which reduces to 1 divided by 13 and in decimal form the answer is a probability of 0.076923076
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Now to your specific problem. 1100 tickets are sold. Your people on your dorm room floor own 187 of them. So when the drawing occurs you have 187 possible winners of the 1100 tickets. The probability of a winning number being held by somebody on your floor is 187 divided by 1100. If you do that division you get an answer of 0.17 as the probability of winning. (This means you have a 17% chance of having the winner be from your floor.)
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Hope this brief description gives you a little more insight into the workings of probability.