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I'm not sure why you included the division by 3. It is not needed.
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You were correct with:
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y + (y-3) + 3(y -3)
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To write the equation you can let P represent the perimeter. Then the equation is:
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P = y + (y-3) + 3(y -3)
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To simplify the right side you can do the distributed multiplication of the last term by multiplying +3 times each of the terms in the parentheses to get:
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P = y + (y-3) + 3y - 9
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Because the remaining set of parentheses (for the second side) is preceded by a + sign, you can just remove the parentheses and you then have:
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P = y + y - 3 + 3y - 9
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Now add up the three terms that contain y ... y + y + 3y = 5y
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and you have P = 5y - 3 - 9
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Finally combine the two constants -3 - 9 = -12
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and you have the final equation:
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P = 5y - 12
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and don't forget that the units for P is yards.
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Hope this helps you to understand the problem better. You had done the hard part correctly except that there was no need to divide the last term by 3. The third side was just three times the second side, and since the second side was y - 3, you only needed to multiply 3(y - 3) to get the third side.
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Keep up the good work ... you're getting it!!!

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The wire is 40 inches long. Therefore, we know the perimeter of the triangle it forms is 40 inches.
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One of the two legs is given as being 15 inches long. Therefore, the other leg added to the hypotenuse have to equal the remainder of the 40 inches. So the unknown leg plus the hypotenuse must be 25 inches (the 40 inches minus the 15 inches equals 25 inches). Let H represent the unknown length of the hypotenuse and let L represent the unknown length of one of the legs. Since the hypotenuse plus the unknown leg must add up to 25 inches we can write:
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We can then subtract L from both sides to get:
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Next, let's look at the Pythagorean theorem. It states that the square of the hypotenuse is equal to the sum of the squares of the two legs of a right triangle. In equation form this is:
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We know that one of the legs is 15 inches long. Let's substitute 15 for one of the legs and this makes the Pythagorean equation become:
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Square the 15 to get 225 and the Pythagorean equation that results is:
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Note that we dropped the subscript on the unknown leg just to simplify the notation.
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Recall that we previously found that H = 25 - L. Let's substitute 25 - L for its equal H in the Pythagorean equation to get:
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Square out the left side of this equation to get:
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We can eliminate the two squared L terms by subtracting from both sides of the equation. The resulting simplified equation then becomes:
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Subtract 625 from both sides of this equation to get:
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Solve for the unknown leg L by dividing both sides of this equation by -50 to find that:
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which reduces to:
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So now we know that the two legs of this right triangle are 8 and 15. Since the perimeter of the triangle is to be 40 inches, the hypotenuse must be the length of the wire remaining. So take the 40 inches and subtract 8 and then subtract 15 and you get 17 inches that is left for the length of the hypotenuse.
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Just to give us some confidence in this answer, return to the Pythagorean equation and substitute these values to ensure that they work for this right triangle.
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Square each of the terms to get:
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If we add the two terms on the right, we do get:
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So our answer checks. The right triangle is a 8 inch, 15 inch , 17 inch triangle.
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Hope that this explanation helps you to understand how this problem, and others like it, can be solved. Good luck!
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Let C represent the reading on the Celsius scale and F represent the reading on the Fahrenheit scale.
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You are then told if you add 3 times C to 2 times F and the result is 460. In equation form write this as:
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Next you are told that if you add 5 times C to 4 times F the result is 860. In equation form this is:
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One of the ways to solve this is by the method of variable elimination. You can eliminate the variable F by multiplying both sides (all terms) of the first equation by 2. When you do that the first equation becomes:
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Write this above the second equation as follows:
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Next, subtract the second equation from the first equation in vertical columns. Note that the 4F in the first equation will be canceled by subtracting the 4F below it in the second equation. This eliminates the variable F. You are left with 6C minus 5C which is just C. And on the right side you are left with 920 minus 860 and this is 60. So what you have left is simply:
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This tells you that the answer for C is 60 degrees Celsius. Go back to either of the two equations and substitute 60 for C. Then solve for F. Since you can choose either equation, you can use:
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Substitute 60 for C and you have:
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Multiply the 5 times 60 to get 300 and the equation becomes:
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Eliminate the 300 from the left side by subtracting 300 from both sides. This subtraction results in:
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which simplifies to:
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Solve for F by dividing both sides of the equation by 4 to get:
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and dividing 4 into 560 results in the answer:
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So the reading on the Fahrenheit thermometer is 140 degrees.
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In summary the answers are C = 60 degrees and F = 140 degrees.
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Hope this helps you to understand the problem better.
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Given to solve for x:
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There are a number of different things we could do to make the problem a little easier. A lot of students have trouble with fractions, so let's begin by getting rid of the fractions. The right side of the equation may cause us a little trouble, so let's begin by doing the distributed multiplication on the right side. Do that by multiplying times each of the two terms in the parentheses to get:
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When you multiply out the last term on the right side [that is when you multiply ] you get which is . So you can now write the equation as:
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Now to get rid of the fractions, we can multiply every term on both sides by the product of all the denominators. That is multiply every term by . (We could use 120, but using 240 does not cause a problem).
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When we multiply by 240 the equation becomes:
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Now in each of the four terms, let's divide the denominator into the 240 in the numerator to get:
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Multiply out all the terms and you have:
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This is in a form that is more like an ordinary algebraic equation. Start by subtracting 96x from both sides:
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Next add 210 to both sides:
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Solve for x by dividing both sides by -16 to get:
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and since both 570 and -16 are divisible by 2 this can be reduced to:
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Or by dividing it out, you can get the decimal equivalent form:
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Hope this helps you to understand the problem a little better. Sometimes it helps to make the problem a little easier by getting rid of the fractions, and you can do that by multiplying all terms by a common denominator (the product of all denominators) and then getting rid of each denominator by dividing it into the common denominator.
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Let C represent the cost of the ride without a tip. So when Greg's family arrived at their destination, the meter said that they owed the driver C.
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They apparently got good service so they decided to add a 20% tip. That means that they multiplied the cost on the meter by 0.2 (0.2 is equivalent to 20%). and they added that to C. When they did that the total was $25.80. In equation form you can write that:
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C + 0.2*C = $25.80
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You can add the two terms on the left side because they both involve C. The 1*C plus 0.2*C equals 1.2*C and this makes the equation become:
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1.2*C = $25.80
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Now you can solve for C by dividing both sides of the equation by 1.2 to get:
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C = $25.80/1.2 = $21.50
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So you now know that C (the cost of the ride) was $21.50. To that amount they added a 20% tip (0.2 times $21.50 = $4.30). So the $21.50 for the ride and the $4.30 for the tip add up to $21.50 + $4.30 = $25.80, just as the problem said it should.
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In summary, the cost of the taxicab ride was $21.50
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Hope this helps you to understand the problem a little better.
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The hardest part of this problem involves understanding the order in which operations (for example: adding, subtracting, multiplying, dividing, taking exponents) are performed. The order can be remembered by PEMDAS. Use this to remind you that you do things in this general order.
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P = do the things inside parentheses first
E = work the exponents second
MD = third, reading from left to right, do the multiplications & divisions as you come to them
AS = finally, reading from left to right, do the additions & subtractions as you come to them.
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For this problem you are asked to simplify:
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-42÷6+(-7)
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This problem involves parentheses containing just the term -7. So we don't have to worry about doing any EMDAS actions inside the parentheses. However, since the parentheses are preceded by a plus sign, we can remove them, eliminate the plus sign, and leave the signs of any terms inside the parentheses unchanged. So the + (-7) becomes just -7 and the expression is simplified to:
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-42÷6-7
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There are no exponents to consider as our second step. So we can proceed to the MD (multiplications and divisions) reading from left to right, we encounter just one division sign. It tells you to divide the number just before it (-42) by the number immediately after it (6). When you do that division the answer is -7. So you can replace -42÷6 by -7 and the problems is then reduced to:
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-7 -7
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There are no more multiplications or divisions to consider. So we can proceed to the AS (additions and subtractions) in the order they are encountered in reading from left to right.
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Reading from left to right we have -7 and from that we subtract 7 (or add -7 to that, if you prefer to view it that way). The result of -7 and -7 is -14 and that is the answer to this problem:
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-42÷6+(-7) = -14
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Hope this helps to provide a little clarification to the order in which algebraic operations are taken in expressions.
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The formula for the area (A) of any triangle is:
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in which B represents the dimension of the Base and H represents the dimension of the Height or Altitude.
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For this problem you are given that B = 100 ft and H = 70 ft. Substituting these two values into the formula results in:
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The 1/2 times 100 results in 50 and then multiplying the 50 times 70 results in 3500. The units are square feet. So the answer you are looking for is 3500 square feet.
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Hope this helps you to understand this problem.
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The minute hand will make one complete rotation in an hour (60 minutes). Picture that at the start of an hour, the minute hand is pointed at the 12 on the clock. It then begins a slow rotation around the clock until 60 minutes later it arrives at the position where it again points to the 12. How far has the tip of the minute hand traveled during that hour? It has traveled the complete circumference of a circle that has a radius of 2 inches. So you can write an equation for this distance as:
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Where C stands for the circumference of the circle and r stands for the radius, and the equation is just the standard equation for finding the circumference of a circle if you know the radius.
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In this problem, the radius is 2 inches and therefore you can substitute 2 for r and you will find the circumference in inches. This is as follows:
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which by multiplying out the right side simplifies to:
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inches
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You can convert this to a more common form by substituting 3.1416 as the value for and multiplies out as follows:
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and this becomes inches per hour.
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So for each hour the tip of the minute hand travels inches or an equivalent numerical value of 12.5664 inches.
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Now to answer the questions that you are asked in the problem. First, how far does the tip of the minute hand travel in 15 minutes? 15 minutes is a quarter (one-fourth) of an hour. So in 15 minutes the tip of the minute hand travels one-fourth of the distance it would travel in an hour. So, using D to represent the distance the tip moves we can say that for 15 minutes the tip moves:
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And dividing out the right side, you get that in 15 minutes the tip of the minute hand moves:
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inches which numerically is inches
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Next, how far does the tip of the minute hand travel in 10 and a half hours? Since each hour it travels inches, in 10 hours it will travel a distance of:
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which multiplies out to inches.
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Then in the additional half hour the tip, since the tip moves inches in an hour, it will move half that distance in the half hour. In other words, in the half hour it moves half of inches, or it moves inches. So in 10 and a half hours it moves the sum of these two distances which is:
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inches and this totals to inches
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As before you can convert this to numerical inches by substituting 3.1416 for to get:
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inches and this multiplies out to inches.
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Hope this helps you to understand how to work clock problems such as this one.
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I'm not sure that this is the best approach, but here's an algebraic way to solve this problem.
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First, note that each change in the series follows an arithmetic pattern. The first term is just the single letter "a". The next group of terms is two of the letter b. The next group of terms are three of the letter c. The next group of terms are four of the letter d. And so on. This means that the total number of letters at any point in the series can be found by adding:
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1 + 2 + 3 + 4 + ........
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Here's a simple check that might clarify this a bit. How many total letters are in the series at the end of the fifth letter which is e? The answer to that is:
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1 + 2 + 3 + 4 + 5 = 15
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And if you count all the letters in the opening of the sample series given in the problem, you find there are a total of 15 letters ... one a plus 2 b + 3 c + 4 d + 5 e = 15.
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In other words, each number in this example series represents a letter. There are 5 numbers (1 through 5) so each number corresponds to a letter (a through e). The value of the number tells how many of each corresponding letter appears in the series. You need to understand this. Maybe the further explanation below will help.
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Notice that this is an arithmetic series. You may recall that the sum (call it S) of the terms in an arithmetic series where each term increases by 1 from the previous term is given by the equation:
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in which n represents the number of terms in the series. And for this problem n REPRESENTS THE NUMBER OF SUCCESSIVE GROUPS OF ALPHABET LETTERS THAT HAVE BEEN USED.
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You need to find out what number n will give you the 280th term, and that can be done by setting the S (for sum) equal to 280 in the equation as follows:
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Multiply both sides by the denominator 2 on the right side and you have:
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Multiply out the right side to get:
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Subtract 560 from both sides:
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And transpose to get the conventional quadratic form of:
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Note that for this problem, you are only interested in positive values for n.
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Apply the quadratic formula to solve for n. The value for "a" in the formula (the multiplier of the n-squared term) is +1. The value for "b" in the formula (the multiplier of the n-term) is also +1. And the value for "c" (the constant) is = -560. Substitute those values into the quadratic formula:
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I'll leave the math work to you. Remember you are only interested in a positive answer for n. You can disregard the negative n.
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After you do the work you should get an answer of n = 23.17
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Think about what this means. You have to go out more than 23 letters to get to the 280th term of the series you were given by the problem. The 23rd letter of the alphabet is w. But you need to go beyond w and into the group of x's. (The end of the x's occurs when n = 24.00). So the answer to this problem is the 280th term is in the group of x's, so the letter x is the 280th term.
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Sort of difficult to explain, but I hope this gives you a way to view how this problem can be solved. And I hope this helps.
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This one might be a little confusing to you.
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Normally, the slope intercept form is:
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in which m, the multiplier of x, is the slope and b is the value on the y-axis where the linear graph of the equation crosses the y-axis.
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The slope is normally computed by dividing the change in x from the two given points into the corresponding change in y. In this case, note that in the two points that are given, x goes from -6.2 in one of the points to - 3.4 in the other. Subtracting the -6.2 from -3.4 results in a difference of +2.8. So in finding the slope, +2.8 is what we'll be dividing into the corresponding change in the y values of the two points.
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But, what is the change in y? The value of y in the first point is 18.1 and the value of y in the second point is also 18.1. The difference in these two values is zero. So while x went 2.8 units toward the right (the run from -6.1 to -3.4) the value of y stayed the same and, therefore, the rise was zero. Dividing 2.8 into zero results in a slope of zero.
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Return to the the slope-intercept form for the equation. When you substitute zero for the slope the equation becomes:
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The multiplication by zero makes the x term disappear and we are left with:
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Now all we have to realize is that y never changes regardless of what value we choose for x. For this problem the value of y is always 18.1 and the graph is a horizontal line (parallel to the x-axis) that crosses the y-axis at +18.1. This means that b is 18.1 and the slope intercept equation is simply:
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That's all there is to it. Whenever you see a slope intercept form that says simply y equals a constant, you should picture a graph that is a horizontal line and crosses the y-axis at the value of the constant.
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Hope this helps you to understand the "trick" in this problem and what it means.
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You can work these problems using the same rules that you have learned for solving equations with this exception: If you multiply or divide both sides of the inequality BY A NEGATIVE Quantity, you must reverse the direction of the inequality sign.
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So, just imagine for the time being that the > sign in this problem is an equal sign.
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You are given:
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You are expected to solve this for x. In an ordinary equation you would get rid of the -5 on the left side by adding 5 to both sides. You do the same here. Just add 5 to both sides and the problem becomes:
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And that's the answer to this problem. x is greater than 5.
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We didn't make use of the exception because we didn't have to multiply or divide both sides by a negative quantity to solve for x. But don't forget that rule. It will be useful for other problems of this type.
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For the time being just think of working these problems by following the rules you already know for equations.
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Hope this helps you to understand inequalities a little better.

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Given to solve:
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I think I'm safe in assuming that by using the term "log", you mean that the base of the logarithms you are using is 10.
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One of the first things you can choose to do is to add +3 to both sides of this equation to get rid of the -3 on the left side and move the constant to the right side as follows:
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Now you have only logarithm terms on the left side. You can apply the rules of logarithms to these terms. Begin by noting that the first logarithm term has the constant 2 multiplying it. By the rules, a multiplying constant can be taken inside the logarithm function as an exponent as shown below:
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There are now two ways that you can deal with the negative logarithm. You can note that when the base of the logarithm is the same as the quantity that the logarithm is operating on, the logarithm term can be replaced by the number 1. Here are some examples:
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; ; ; and
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So you could immediately replace by and you would have:
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Then you could add 1 to both sides to get rid of the -1 on the left side. The result would be:
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And you could proceed to solve for x from this point by using the conversion from logarithmic form to exponential form that I'm going to describe a little bit further on.
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But instead of doing it that way let's do it another way that contains the lesson that the difference between two logs can be expressed as the logarithm of their division in which the denominator is the quantity that the negative logarithm term is operating on. Sounds hard, but it's pretty easy actually. Going back to the point we had arrived at, which was:
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The rules of logarithms say that you can convert the two terms on the left to a single logarithm in which the quantity from the first logarithm is divided by the quantity from the negative logarithm as follows:
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Next we're going to take a look at converting logarithms to exponential form. This is an important property to learn because so many logarithm problems (like this problem) make use of it. The conversion equation says that by the definition of logarithm:
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is the same as saying
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This says "If you have a logarithm form, you can change it to an exponential form by taking the base of the logarithm, raising it to the exponent of the quantity on the other side of the equal sign, and setting that result equal to the quantity that the logarithm is operating on." Let's continue. We have arrived at:
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Let's convert this to exponential form. Take the base (which is 10); raise it to the power of the 3 on the right side of this equation, and set that equal to . This leads to the exponential equation:
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I hope that you followed that. Now get rid of the denominator 10 by multiplying both sides of this equation by 10. On the left side multiplying
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and by adding the exponent 3 to the exponent 1 implied for just the 10) this multiplication results in:
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and on the right side, the multiplication by 10 eliminates the 10 in the denominator. So this equation has been reduced to:
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Now you can solve for x simply by taking the square root of both sides. When you do that the exponent on the left side is just divided by 2 to give you the square root (think . And on the right side the x-squared becomes just x. So we have the solution:
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And the 10 squared is just 10 times 10 which is 100.
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So the answer to this problem is x = 100.
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Hope that this long explanation helps you to understand logarithms a little better. It's just a matter of practice and thinking about all this. It'll become easier with working on these types of problems until you get familiar with all the rules of logarithms. Keep working at it.
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Given to simplify:
.

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Look at the first term, specifically look inside the parentheses. The parentheses contain the difference of two logarithms and by the rules governing logarithms, the difference of the logarithms of two quantities is equal to the logarithm of the division as shown in the following:
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Next by the rules of logarithms, the 2 that multiplies the logarithm of the simplified first term can become the exponent of the quantity that the logarithm is operating on. When this rule is applied, the first term changes as shown in the following expression:
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Still operating on the first term, use the power rule of exponents to square the quantity that the log function is operating on and get:
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Moving on to the second term in the expression, change the 2 that multiplies log(5) so that it becomes the exponent of 5 as in the following:
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and square the 5 to get:
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Now recognize that the sum of two logarithms equals the log of the product of the quantities that the two logarithms are operating on:
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Multiply out the 3*25 to get 75 so that the expression now is:
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You again have the difference of two logarithms which means that by the rules of logarithms you can change this to the logarithm of the division of the two quantities that the logarithms are operating on:
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Dividing by 75 is the same as multiplying by 1/75 and this gives you the final simplified answer of:
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Hope this helps you to understand the problem. Make sure you understand each of the rules for logarithms that are applied in solving this problem.
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A logical first thing to do in solving this problem is to determine in which of the four Cartesian quadrants the angle theta lies. Do this by examining the signs of the two trigonometric functions that you are given for theta.
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First, notice that you are told that sin (theta) is greater than zero, meaning that sin (theta) is positive. (Think of the definition of the sine function as being the side opposite theta divided by the hypotenuse.) All trigonometric functions are positive in Quadrant I. The sine is also positive in Quadrant II where the side opposite is positive (above the x-axis), and the hypotenuse (the side that has unit length and rotates about the origin) is always considered positive regardless of the quadrant. Therefore, by being positive only in Quadrants I and II the sine (side opposite divided by hypotenuse) limits theta to Quadrants I and II and consequently theta will be between 0 and 180 degrees (equivalent to 0 to pi radians).
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Next notice that you are told that tan (theta) is negative. (Recall that the tangent function is defined as the side opposite to the angle divided by the side adjacent to the angle.) To be negative the tangent must have the side opposite to the angle and the side adjacent to the the angle be of opposite signs. Recall that in Quadrant IV, the side opposite the angle will be negative (below the x-axis) and the side adjacent to the angle will be on the positive x-axis. Therefore, the tangent will be negative in Quadrant IV. Next think about Quadrant II. The side opposite the angle will be positive (above the x-axis) and the side adjacent to the angle will be on the negative x-axis. Therefore, the tangent will be negative in Quadrant II.
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So, we have the positive sine value limiting the answer to Quadrants I and II and the negative tangent value limiting the answer to Quadrants II and IV. Theta must be in Quadrant II because that is the only Quadrant that meets both criteria, positive sine and negative tangent.
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Now that we have determined the Quadrant in which theta is positioned, we can move on by working to find where the rotating hypotenuse is positioned. We have been given that:
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Dividing -12 by 5 converts the value of the tangent to -2.4 so we can change the equation for the tangent to:
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That being the case, we can use a scientific calculator to look up the angle that has -2.4 as its tangent. The method to do this will vary depending on the calculator, but what we are going to use is the function which should be read as "the angle whose tangent is x." In our case, x is -2.4. So on a typical scientific calculator set for the degrees mode, you enter -2.4 and press the tan^-1 function to find that the answer is (in degrees) -67.3801. In Quadrant II this angle is measured from the negative y-axis clockwise to the hypotenuse which is the rotating line of unit length and with one of its ends at the origin.
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But theta is measured from the positive x-axis counter-clockwise (or if you prefer anti-clockwise) to the hypotenuse. That means that theta is an obtuse angle equal to 180 degrees minus the 67.3801 degrees between the negative x axis and the hypotenuse. This results in theta being equal to 112.6199 degrees. And that's the answer you are looking for.
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If you want, you can convert this angle to radians by solving for x in the proportion that reads 112.6199 degrees is to 360 degrees as x radians is to 2*pi radians. You should find that the answer for theta in radians is 1.9656.
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Check by again ensuring that your calculator is in the degrees mode. Then clear and enter 112.6119 and press the tan key. Your answer should be -2.399995864 and this rounds to -2.4. This is the correct value, equivalent to -12/5. You can clear this and again enter 112.6199. This time press the sin key to find that the sine of 112.6199 degrees is +0.923076687. Although we were not given a required value for the sine function, this calculation verifies that the sine of an angle of 112.6199 degrees is positive as it was required to be.
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Hope this helps you to see your way through this problem. The method is fundamental to understanding the relationship of trig functions to the Cartesian coordinate system, so you should get an understanding of what is involved in solving this problem because it will help you considerably in working similar problems.
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Given to solve:
.

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Add x to both sides to eliminate the -x on the right side. When you do that, the inequality becomes:
.

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Subtract 2 from both sides to eliminate the +2 on the left side. You get:
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Divide both sides by 2 to solve for the limits on x:
.

.
And that's the answer.
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Note that you can work these inequalities just as you would an equation, with the exception that if you divide or multiply by a negative quantity you must reverse the direction of the inequality sign. This exception did not apply in this problem that you were given to solve.
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Hope this helps you to better understand how to work inequality problems.
.

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You said:
.
ST is congruent to TU: Given <--- Good
V is the midpoint of SU: Given <--- Good
SV is congruent to VU: Def. Of midpoint <--- Good
I'm stuck on what to do next.
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If you can prove part A) triangle SVT is congruent to triangle UVT, then proving B), C), and D) can be done using the properties of congruent triangles.
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There are several ways you can prove part A). Here's one:
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You have already said ST and TU are congruent.
Furthermore, you have correctly said SV is congruent to VU by definition of midpoint.
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Now note that TV is a common side in the triangles STV and UTV. And TV is congruent to TV: Identity
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All you need to do now is say that triangle STV is congruent to triangle UTV using side-side-side congruent to side-side-side.
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Note that you can now say that D) is true, Angle S is congruent to Angle U because corresponding parts of congruent triangles are congruent.
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Or another way you could have proven A) is to also to begin with the statements you have already made:
.
ST congruent to UT: Given
V is the midpoint of SU: Given
SV is congruent to VU: Def. Of midpoint
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Then add the following (presuming you already have studied these properties):
Triangle STV is isoseles: Given
You have already said ST congruent to UT: Given. So you can add that
Angle S congruent to Angle U: Angles opposite of congruent sides in isosceles triangles are congruent. [However, this is item D) you are to prove and you are just stating a reason for item D). So probably the best way to do this problem is to use the side-side-side method and then use the results of that to prove D).]
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And now you can say triangle STV is congruent to triangle UTV using side-angle- side congruent to side-angle-side using ST-Angle S-SV congruent to UT-Angle U-UV
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Now that you have demonstrated A) you can go on to B) TV is perpendicular to SU by noting the following using appropriate words that your instructor will accept:
.
SU is a straight line (180 degrees)
Angle SVU congruent to Angle UVT:Corresponding parts of congruent triangles
mAngle SVU + mAngle UVT = 180 degrees
Substitute mAngle SVU for its congruent counterpart mAngle UVT to get:
mAngle SVU + mAngle SVU = 180
2 * mAngle SVU = 180
mAngle SVU = 180/2 = 90
If mAngle SVU = 90 then its congruent mAngle UVT also equals 90
Therefore TV perpendicular to SU: definition of perpendicular.
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On to C)TV bisects Angle STU.
.
Angle STV is congruent to Angle UTV: Corresponding parts of congruent triangles
Therefore, TV bisects Angle STU: definition of angle bisector.
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And finally D)Angle S is congruent to Angle U.
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If you used the side-side-side congruent to side-side-side procedure for showing that triangle STV is congruent to triangle UTV then you can immediately say that Angle S is congruent to Angle U because corresponding parts (both angles and sides) of congruent triangles are congruent.
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Hope this gives you enough information so that you can wade your way through this problem. You were off to a very good start with what you had done already.
.

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Given the equation:
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You are to find the slope of the graph for this equation.
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One way of doing this is to rearrange the given equation so it is in the slope-intercept form which is:
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When you get it into this form, m, the quantity multiplying the x term, will be the slope, and b will be the value on the y-axis where the graph crosses.
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Start with:
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and the plan is to solve for y with everything else on the right side.
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Start by subtracting 3x from both sides. When you do that the 3x disappears on the left side and the equation becomes:
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Next solve for y by dividing both sides (all terms) by 7 to get:
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This is in the slope intercept form. You can see that the multiplier of the x term is and that is the value of the slope. The b term is and that is the value on the y-axis where the graph crosses.
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So, the slope is and that is the answer to this problem.
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Hope this helps you to understand this problem.
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You are asked to evaluate
.

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when x = 2 and y = -4
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Start by evaluating
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When x = 2 then . This is equal to
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Next evaluate
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Substitute 2 for x and -4 for y to get:
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. The 3*2 = 6 and then the 6*-4 = -24
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Finally, evaluate when y = -4. Substitute -4 for y to get:
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and this becomes
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Now combine all three terms. Don't forget that this last one has a minus sign preceding it.
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Remove the parentheses and then sum the terms on the right side to get the answer:
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Combine the +8 and -24 to get -16. Then add the -16 to the -64 to get -80.
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Hope this helps you to find where you were having trouble. In this case it could easily involve being confused by the signs.
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We were given the equation:
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First, we need to find the slope of the graphed line that represents this equation. To do that, let's transform the given equation into the slope-intercept form
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Subtract 4x from both sides of the given equation. The result is:
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Solve for y by dividing both sides (all terms) by 6
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Note that the fraction (4/6) reduces to (2/3) when both the numerator and denominator are divided by 2. This makes the slope-intercept form become:
.

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Here are the two critical things to this problem:
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First, in the slope-intercept form of an equation, the multiplier of the x (called the coefficient of x) is the slope of the graphed line for that equation.
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And second, any line perpendicular to the graphed line has a slope that is the negative and inverse of the slope for the graphed line.
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So looking at the equation that we have in the slope-intercept form, we see that the slope for its graph is because that is the multiplier of the x. (This -2/3 slope means that for every 3 units you move horizontally to the right in the x direction, the graph goes down 2 units in the y direction.)
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Next we know that a line that is perpendicular to it will have a slope that is the negative of this (means change in sign) and is the inverse in value. (An inverse of an integer, means just put that integer as a denominator under a numerator of 1. So, for example, the inverse of 5 is . For fractions, such as we have, we can quickly find the inverse by flipping the fraction upside down.) Applying these two characteristics, we see that the inverse of is and then taking the negative of that changes it to positive. So we found that the slope of the line perpendicular to our given equation is and it is the negative inverse of the slope of the given equation.
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Let's put that into a slope-intercept form for the new equation we are getting for the perpendicular.
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Again the basic slope intercept form is:
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In which m (the multiplier of x) is the slope we want and b is the point where the graph for this equation crosses or intercepts the y-axis. So let's substitute the slope that we want into this equation to make it:
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Next we make use of the fact that we want the graph of this equation to go through the given point (7, -6).
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We know that the x value 7 and its corresponding y value -6 have to work in our equation, meaning that they have to be on the graphed line for this new equation. So we start with:
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and substitute 7 for x and -6 for y to get:
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Multiply out the fraction on the right side to get:
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Solve for b by subtracting 21/2 from both sides as follows:
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and is the same as so we can substitute that to get:
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and combining the two fractions on the left side to get:
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Now we can return to the equation we are finding:
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and substitute -33/2 for b to get:
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and that's the answer: the equation of the answer in slope intercept form for the perpendicular to the graph of the given equation.
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We can convert this answer to the same form of the equation that we were given in the original problem. Begin by subtracting from both sides to get:
.

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Then you can get rid of the denominator 2 by multiplying all term on both sides of this equation by 2 to get:
.

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The graph of these two equations, the original equation and the equation for the perpendicular to it that goes through (7, -6) is shown below:
.

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The red graph is for the original equation we were given, and the green graph is the graph of the equation that we developed for its perpendicular. As a check, you should be able to see that the point (7, -6) is on the green graph of the perpendicular.
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Hope this helps you to understand how to find perpendiculars to the graphs of equations that you are given.
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There are 36 combinations (six outcomes on the first die times 6 outcomes on the second die) that can be rolled when you roll a two dice.
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How many different combinations that total to 9 can be rolled? (First die 6, second die 3); (First die 3, second die 6); (First die 5, second die 4); (First die 4, second die 5). So there are 4 possible combinations totaling 9.
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How many different combinations that total to 10 can be rolled? (First die 6, second die 4); (First die 4, second die 6); (Both dies 5). So there are 3 possible combinations totaling 9.
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This means that there are a total of 7 chances of winning (4 from a 9 and 3 from a 10) by scoring a 9 or a 10 on the first roll.
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Therefore, the chance of winning on the first roll is 7 in 36 or the probability is 7 divided by 36 and this is 0.1944 which is expressed as a probability of 19.44%.
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Hope this helps you to understand the problem. And to help you with future problems involving a dice pairs, here are the possibilities for each roll:
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Rolling a 2: one chance (1&1)
Rolling a 3: two chances (2&1)(1&2)
Rolling a 4: three chances (3&1)(1&3)(2&2)
Rolling a 5: four chances (4&1)(1&4)(3&2)(2&3)
Rolling a 6: five chances (5&1)(1&5)(4&2)(2&4)(2&2)
Rolling a 7: six chances (6&1)(1&6)(5&2)(2&5)(4&3)(3&4)
Rolling an 8: five chances (6&2)(2&5)(5&3)(3&5)(4&4)
Rolling a 9: four chances (6&3)(3&6)(5&4)(4&5)
Rolling a 10: three chances (6&4)(4&6)(5&5)
Rolling an 11: two chances (6&5)(5&6)
Rolling a 12: one chance (6,6)
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The sum of the numbers of possible outcomes is: 1+2+3+4+5+6+5+4+3+2+1 = 36

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The equation for the distance a body has fallen in time t is:
.

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In which t is the total time in seconds that has elapsed since the body started to fall, g represents the acceleration due to gravity (usually in meters per second or feet per second), and D is the total distance that the body has fallen at time t (in meters or in feet, depending on the units used for g).
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Here's a little test-taking strategy. This problem can be solved by using actual time intervals, and this way of solving it is likely to be faster and easier than solving the problem by involving the nth time interval. The possible answers that are given tell you that you can select any two consecutive intervals and the results will be the same.
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The problem tells you to use x for the first interval of 1 second and y for the next interval of 1 second.
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So let's just use the first second as the time interval from t = 0 to t = 1 second. At t = 0 the body has not begun to fall, so the distance it has fallen is zero. By t = 1 second the body has fallen a distance of:
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Notice that the time that the distance the body is down at one second is found by substituting 1 for t in the equation. Multiplying everything out on the right side of this equation results in:
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And the distance the body has fallen during this first second if found by subtracting its distance at one second from its distance at zero seconds (which we know is zero) and we get for x (the distance fallen in the first second) is
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This simplifies to just:
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Next we solve for y (the distance that the body falls during the interval from 1 second to 2 seconds). We already know that at the start of this interval (that is at time t = 1 second) the position of the body is:
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At the time t = 2 seconds the position of the body can be found by substituting 2 for t in the distance equation to get:
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We square the time in the numerator (to get 4) and divide it by the 2 in the denominator the result is that at 2 seconds the body has fallen a total distance of:
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Now all we have to do to solve for y (the distance fallen in the second time interval) is to take this distance at time equals 2 and subtract from it the distance at time equals 1 (which we already found was g/2) and we have:
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and this simplifies by subtracting these two terms to give:
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So at this point we have and
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Next all we have to do is substitute these two values into the possible answers and find out which answer is correct.
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Obviously x does not equal y, so the last answer in the list (x = y) can be eliminated easily.
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How about the first answer on the list,
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Substituting for x and y that would give
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The answer is on the right side of this equation. Dividing the denominator (2) into the numerator (4g) is 2g not g, so that doesn't work and cannot be correct.
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The next answer is
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This time, substituting for x and y would give
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The answer that results from dividing the denominator (2) into the numerator (-2g) is -g not +g, so that doesn't work and cannot be correct.
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Finally, let's try the next answer
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And substituting for y and for x results in:
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The answer that results from dividing the denominator (2) into the numerator (2g) is +g, so this answer does work and, therefore, is the correct answer.
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Hope this helps you see a way how this problem can be done. If you feel that your teacher wants to see it done for any two consecutive time intervals in general, here is what you need to do:
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Use t = n for the start of the first interval. This will give you at the start of this interval.
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Next use t = n + 1 for the end of the first interval (as well as for the start of the second interval). The result will be that the distance at the end of the first interval is given by:
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and this equals . This becomes:
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So the distance X that is traveled between t = n and t = n + 1 is the difference between the distances at these two times as follows:
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and this becomes
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Then use t = n + 2 for the end of the next interval. Substitute n + 2 into the distance equation to find that at time n + 2 seconds the total distance traveled at n + 2 seconds is:
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and that expands to which becomes:
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Then subtract from that answer the answer you got for the distance at the end of the first interval, namely the answer for the equation:
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The result of this subtraction will be y as follows:
.

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And this simplifies to:
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Recall that the answer we found to be correct by doing it the first way was y – x = g
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Let’s try that answer again using these new results and
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Substituting for y and x results in:
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And this simplifies to:
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which becomes
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This is the more general way of showing that the correct answer is y – x = g
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Check my math in all of this just to ensure that I didn't make some mistakes in signs or something else equally as careless.

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Given to graph:
.

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Rearrange the terms containing x and the terms containing y as follows:
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Add 6 to both sides to get rid of the -6 on the left side. Doing this results in:
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Add +1 and +9 to both sides:
.

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You got the +1 by dividing the coefficient of the x term (+2) by 2 to get +1 and then squared the +1 to get +1 which you then added to both sides. You got the +9 by dividing the coefficient of the y term (-6) by 2 to get -3 and then squared that to get +9 which you then added to both sides.
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The left side now contains two tri-nomials, one in x and one in y as grouped below:
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Note that on the each of the two groups on the left side are perfect squares as shown:
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Add the three constants on the right side:
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Note that the constant on the right side is equal to . Substitute this in for the right side and you have:
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This should now be a form that you recognize. It is a circle centered at (-1,3) and having a radius equal to 4.
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Hope this helps you to understand a method of working with equations such as this. We just did some additions of constants to form two perfect squares ... using the method for completing the square as you probably did at one time when you were learning how the quadratic formula was developed.
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You were on the right track.
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Given to factor:
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Separate the first three terms as follows:
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Note that the terms in the parentheses are an algebraic expression that is a perfect square. It can be factored as shown below:
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Notice now that this new form is the difference between two squares. Recall the rule for the difference of two squares:
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For your problem and
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Substitute these into the rule for the difference of two squares as shown in the steps that follow:
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.

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Remove the interior parentheses on the right side of this equation and you have:
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The right side is the factored form of the problem that you were given. So you can write the answer to this problem in the equation form:
.

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or maybe you want to rearrange the terms in the parentheses a little so the letters come first as in:
.

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Whatever your teacher prefers. It's just a matter of preference, but the rearrangement is probably the most commonly used form.
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Hope this helps you understand the problem a little better.
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Thanks for helping your daughter with math.
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The hardest part of this problem involves translating units. You were correct in your initial plan to divide 200 by 6. The 200 is in units of yards and the 6 is in units of seconds. So when you do that division, your answer will have units of yards per second. But the problem asks for the answer in miles per hour and here the "fun" begins. How do you get from yards per second (which means units of yards divided by units of seconds) to miles per hour?
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The process that I'm about to describe will seem complex, but with a little thought and practice, it's not too bad. The big advantage is that it helps to show you what you need to do, and it helps to ensure that you don't mess up on the conversions that you need to do.
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Start by writing down the units at the beginning. In this case you are beginning with yards per second as follows:
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Next, write down the units that you want to get to. In this problem it is miles per hour as follows:
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Some how we need to convert yards into miles and seconds into hours. There are 1760 yards in a mile. To cancel out the yards units in the cheetah's speed, we need to multiply in by a units factor that has yards in the denominator and the corresponding miles in the numerator. So going back to the cheetah's speed in yards per second we can multiply it by the conversion factor as below:
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Notice that after we do that multiplication the unit yards in the numerator will cancel with the units yards in the denominator and we will be left with units of miles per second.
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Now we need to get rid of the seconds by converting it to hours. In our units equation we have seconds in the denominator so we need a multiplier to cancel it out by having seconds in the numerator. There are 3600 seconds per hour (60 seconds in a minute times 60 minutes in an hour equals 3600 seconds per hour.)
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So in the conversion arrangement that we have started, let's add another multiplier that will cancel the seconds and introduce the hours. This is shown below:
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Now let's add to this arrangement the conversion numbers. The cheetah's speed is 200 yards in 6 seconds. Put that in:
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Next, 1 mile is 1760 yards. Put that in:
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Finally 3600 seconds is 1 hour. Put that in:
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Cross out any units that appear in both the numerator and the denominator:
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Note that the only units that are left are miles and hour. Now all we have to do is multiply the numbers in the numerator (200 times 1 times 3600) and divide that answer by the numbers multiplied in the denominator (6 times 1760 times 1). With the cancellation of the units, you are left with:
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The numerator becomes
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And the denominator becomes
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Divide the 720,000 by 10,560 and you get 68.1818 miles/hour. That's about the right speed for a cheetah. They can sprint around 70 miles per hour, so our answer is around what it should be.
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If your daughter takes physics in high school, she will be involved with a lot of problems that have unit conversions such as this one. This method of working conversions will come in handy and help to do the conversions correctly.
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I hope this wasn't too confusing. As I said, with a little thought and practice it gets easier.
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Good luck and thanks again for helping your daughter. Hopefully this site will assist you in doing that for her.

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One piece will be length x and the other piece will be 20 cm longer, so it will be x + 20 cm long.
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Added together the length of these two boards will equal 86 cm. So you can write an equation:
.
x + (x + 20) = 86
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Remove the parentheses and add the two x's together to get:
.
2x + 20 = 86
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Subtract 20 from both sides:
.
2x = 66
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Divide both sides by 2 and you have:
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x = 33
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The short piece is 33 cm and the other piece is 20 cm longer or 33 + 20 = 53 cm.
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As a check you can add 33 + 53 to make sure that the lengths of these two pieces add up to be the total length of the original board, 86 cm. They do, so your answer is correct.
.

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Lots of room for making mistakes in this one.
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First note that the volume is in cubic cm and the surface area is in square inches. We need to be consistent in units, so let's convert the volume to cubic inches. I know that 2.54 cm = 1 inch. By cubing both sides of this equation we get:
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cu cm and this equals 1 cu in.
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So by dividing 196 cu cm by 16.387 cu cm per cu inch we get that the volume is 11.96 cu inches. For simplicity, let's just call it 12 cu inches.
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Next let's compute the surface area of the box in terms of the unknown dimensions of L (for length), W (for width), and H (for height).
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First the area of the bottom of the box is L*W. And the top (or lid) of the box has the same area, L*W. So the combined surface area of the top and the bottom is 2*L*W.
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Now for the area of the long sides of the box. There are two long sides. Each of them has a dimension of L and it is multiplied by the height to get its surface area. Since there are two of these sides, the combined surface area of these two sides is 2*L*H.
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And finally for the area of the short sides of the box. There are two of these. Each of them has a dimension W which gets multiplied by the height (H) to find its surface area. For the two of these sides the surface area is 2*W*H.
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Now add all these surface areas to get the surface area of the total box.
.

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But the problem tells us that L = 2*W. Substituting 2*W for L in this surface area equation gives:
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Multiply out each term to get:
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Add the two terms containing W*H and the equation becomes:
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Finally, we can simplify this by dividing all terms on both sides by 2 to get:
.

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This is our first equation.
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Next we can look at the second equation. The plan is to solve it for H in terms of W and then substitute that equation for H into the first equation. This will make the first equation have only the variable W, and we can then solve for W.
Our second equation is for the Volume (V). The equation for the volume of a box is:
.

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We were told that the volume was 196 cu cm, which we converted to cu inches. We found that this box was 12 cu inches. So substitute 12 for V. Also we were told that L = 2*W. So also substitute 2*W for L in the volume equation:
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Multiply out the right side:
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Divide both sides of this equation by 2, and then divide both sides by to get:
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Next we can substitute for H in the first equation and we have:
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If we multiply out the second term on the left side, and cancel the W in the numerator with one of the W's in the denominator this equation becomes:
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Simplify this by dividing the entire equation (all terms) by 2 and the result is:
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Then get rid of the denominator in the second term on the left side by multiplying both sides (all terms) by W. We get:
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Subtract 70W from both sides to put the equation in the more standard form of:
.

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Now all that has to be done is to solve this equation for W. This equation does not factor nicely. One way we can get a fairly good approximation of the answer is to define w = x and then use a graphing calculator to graph the equation:
.

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The graph crosses the x-axis at three points. (x-axis crossings are where y = 0, so the value of x (or W) will be the value that will make the equation equal zero.) One of the crossing points was at a negative value of x (or W). We can eliminate that because a negative width for the box makes no sense. The other two places that the graph crossed the x-axis gave a positive value for x or W. One was between 0 and 1 on the axis, and the other was between 8 and 9. By zooming in and tracing down the curve and then just trying logical values for W I was finally able to find that values for W of about 0.129 and 8.302 made the equation work so that:
.
came within a few hundredths of equaling zero.
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For W = 0.129 inches the corresponding value of L (which is 2*W) is 0.258 inches and the corresponding value of H (which is 6/W^2) is 360.555 inches. (This box is a tower that is a little more than 30 ft in height. Not too practical.)
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For W = 8.302 inches the corresponding value of L (which is 2*W) is 16.604 inches and the corresponding value of H (which is 6/W^2) is 0.087 inches. (This box has a reasonable length and width, but it is very thin ... less than a tenth of an inch high.)
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There are two possible boxes that have dimensions that will make the volume approximately 196 cu cm (12 cu inches) and the surface area equal 280 sq inches. As a check, you can use the above approximate dimensions to show that the volume and surface area for each set of dimensions are 12 cu inches and 280 square inches respectively.

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You are looking for the speed (or rate). Use the equation that says:
.

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Substitute D for distance, R for rate, and T for time to get:
.

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Does this equation make sense? Sure it does. Suppose your rate or speed while driving a car is 60 miles per hour, meaning that every hour you drive you go 60 more miles. If the time (T) that you spent driving was 3 hours, then the distance you drove is 60 * 3 = 180 miles and that's what the equation tells you to do ... multiply your speed times your driving time to get the distance you went.
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Now let's apply this formula to the problem that you were given. The plane flew a distance (D) of 280 miles. The time it took (T) was 70 minutes. Substitute these two values into this formula and you have:
.

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Solve for the speed R by dividing both sides by 70 to get:
.

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Do the division on the left side and the answer becomes:
.

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So the speed R is 4. What are the units of this speed? Well, the 280 is in units of miles and the 70 is in units of minutes. Therefore, the speed or rate is in units of miles per minute.
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The plane is flying at a rate of 4 miles per minute. Every minute that passes, the plane goes another 4 miles. So in 60 minutes (one hour) at the speed of 4 miles per minute, the plane goes 4 times 60 or 240 miles. Therefore, its speed in miles per hour (a more common unit for airplane speed than miles per minute) is 240 miles per hour.
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Since the problem didn't ask you for any particular units for the speed, you could answer 4 miles per minute or 240 miles per hour. Either answer is correct.
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Hope this helps you to think your way through this problem and understand it. Remember the equation . It is useful in many problems involving speed, distance, and time.
.

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Consecutive odd integers are two units apart. Example 3 & 5 or 101 & 103.
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That being the case, if n represents the first positive odd integer, then n + 2 represents the next positive integer. n +2 is the larger odd integer.
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The problem tells you that the first of these two integers (that means n) is to be squared. Squaring n results in
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Next you are told that this quantity is to be equal to "4 more than 5 times the larger." Well then, 5 times the larger is 5*(n + 2) and 4 more than that is 4 + 5*(n + 2). This is to equal n squared. So we can write the equation:
.

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Multiply out the 5*(n + 2) on the right side. It becomes 5n + 10. So this makes the equation:
.

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Add the 4 and the 10 on the right side to get:
.

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This is a quadratic relationship. Put it into standard quadratic form by subtracting 5n + 14 from both sides to get:
.

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The left side of this equation can be factored as follows:
.

.
and this equation will be true whenever one of the factors equals zero. This is because multiplication by a zero on the left side makes the entire left side equal to the zero on the right side. This means there are two possible solutions. Either:
.
which means that
.
or
.
which means that
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But the problem says that n must be positive and odd. The only answer that satisfies both those conditions is .
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So if then the next consecutive odd number is .
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The problem asks for the sum of these two digits so the answer is
.
Let's just check to make sure that the two digits satisfy the conditions of the problem.
.
and that is
.
and that also equals
.
Therefore, we can say that we worked it out correctly. We have the two correct positive odd integers, and their sum (which the problem asked us to find) is 16.
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Hope this helps you to understand how to work this problem and gives you some insight into how to solve similar problems.
.

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There is NOT enough information for a single answer. Any odd number greater than 30 will satisfy the two criteria (odd and greater than 30).
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For example 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, ... are each odd and greater than 30, so any one of them could be the number you were thinking of.
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You are given to solve for x:
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It always helps me to put the radicals on different sides of the equation. In this case let's subtract from both sides to make the equation become:
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Now square both sides:
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When you square a square root, the answer is just the term under the radical sign. So the left side changes as shown below:
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We can square the right side by doing a FOIL multiplication as shown:
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The FOIL multiplication results in:
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On the right side combine the two radical terms. Then group the two non-radical terms. This will give you:
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Get all the non-radical terms on one side of the equation. You can do this by subtracting 25 and 4x from both sides. This leaves the radical by itself on the right side and the left side becomes:
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Combine the like terms on the left side (5 and -25 = -20, and -x and -4x = -5x) to get:
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Since all the terms are negative let's multiply both sides (all terms) by -1 to change the equation to:
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Now square both sides again. This involves another FOIL multiplication on the left side and squaring a radical (and the 10) on the right side so that you have:
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The FOIL multiplication results in:
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Combine the two x terms on the left side and multiply out the right side to get:
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Subtract 400x from both sides to get rid of the 400x on the right side:
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Simplify by dividing all terms on both sides by 25 to get:
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Rearrange the terms on the left side in descending order of the power of x:
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The left side factors:
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To make the left side equal to the right side it will require that x-4 = 0. Solving this (by adding 4 to both sides) tells you that x = +4 is the solution to this problem.
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You can check this solution by substituting +4 for x in the original problem:
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becomes:
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which becomes:
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and this simplifies to:
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It checks. And we can say with confidence that x = 4 is the solution to this problem.
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Hope this helps you to understand the problem and helps you to see how to multiply out the radicals.
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You can put this solution on YOUR website!
Given to solve:
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You first need to get this into the standard quadratic form of:
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Do this by getting rid of the 6 on the right hand side. To make that happen, subtract 6 from both sides to get:
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On the left side the 15 and -6 combine to +9. And on the right side the 6 and -6 combine to zero. This changes the problem into:
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Note that this is in the standard quadratic form where "a" (which is the multiplier of the term) is equal to 1, b (which is the multiplier of the term) is -6, and c (the constant term) is +9.
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Quadratic equations are generally solved using one of three methods:
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a. factoring
b. completing the square
c. using the quadratic formula
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This problem can be solved by factoring. You can factor the left side into:
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Note that the two factors are the same. If you FOIL multiply these two factors you will get
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The factored equation will be true if either one of the two factors is equal to zero because a multiplication by zero on the left side makes the entire left side equal to zero which is equal to the right side.
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You can solve for the value or values of x that will make the factor equal to zero by setting each of the factors equal to zero and solving for x. This is as follows:
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add 3 to both sides to get rid of the -3 on the left side and this becomes:
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Since the other factor is the same, it also will go to zero if x = 3.
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So the solution to this problem is x = 3.
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Check by substitution 3 for x in the equation that you were given in the problem. Start with:
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Substitute 3 for x to get:
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Square the 3 and the equation becomes:
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Multiply the -6 times 3 to get -18 and the equation is then:
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Algebraically sum the left side. 9 + 15 = 24 and subtract 18 to get 6 on the left side. This equals the right side, so the answer checks. When x equals +3 the left side of the equation that you were originally given becomes equal to the right side.
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Hope this helps you to understand the problem. From lots of experience I was able to see that the left side would factor. However, not all quadratic equations can be factored. If you are unable to factor the left side, you can always use the quadratic formula to solve the quadratic equation. (We could have used the quadratic formula on this problem.) Using the quadratic formula is a whole different way of finding the answer and generally takes more work than factoring.