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I think this is what the problem is looking for you to do:
.
At the low end the temperature has always been equal to or greater than -54 degrees F.
.
At the high end the temperature has always been equal to or less than 119 degrees F.
.
So the temperature has always been in the range from -54 to +119 degrees Fahrenheit
(inclusive -- meaning that the range includes -54 and +119).
.
You can express this in the form of a trichotomy inequality:
.

.
where the numbers are understood to be in units of degrees Fahrenheit and this
inequality
says that the temperature has always been between the two limits ... bigger than or equal
to -54 and less than or equal to +119.
.
Hope this helps you to understand the concept of range.
.

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You are given:
.

.
To simplify this you can use the rule that says:
.

.
Applying that rule to we can break the 50 up into the product of 25*2.
So we can say that
.
But . So we can substitute 5 for and the problem simplifies
further:
.

.
Now let's do the same thing for . Factor 18 so that you have:
.

.
And so this part of the problem simplifies further to:
.

.
Put the two parts of the problem together ...
.

.
If you factor from the two terms on the right side you get:
.

.
So the answer to this problem is:
.

.
Hope this helps you to understand one of the rules that can be used in simplifying
radicals.
.

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Let X represent the total number of points on the test.
.
The problem tells you that 88% of X equals 66 points
.
Change 88% to its equivalent decimal form of 0.88.
.
Then multiply 0.88 times X and set that equal to 66 points. In equation form this is:
.
0.88X = 66
.
Solve for X by dividing both sides of this equation by 0.88 and you get:
.

.
Calculator time ... when you divide 66 by 0.88 you get 75. So the answer to this problem is:
.

.
There are 75 points on the exam.
.
Check: If the test has 75 points and you got 88% (or 0.88) of the points correct then you
scored 0.88 times 75 which equals 66 points. This checks with the information in the problem.
.
Hope this helps you to understand percent a little more.
.

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Chama ---
.
You probably can just about do these problems in your head once you understand what is going on.
.
Begin by thinking about the coordinate system of axes. A point on the y-axis must have a corresponding
x value of zero. For example, what is the coordinate pair that corresponds to the point +5
on the y-axis. The pair is (0, +5). That means that for a given linear function if you set x
equal to zero, y will equal the intercept value on the y-axis.
.
Similarly, any point on the x-axis will have zero for its y value. Example, the point (-3, 0)
is on the x-axis. So if you set y equal to zero in the given function, the corresponding
value of x will be on the x-axis.
.
Think about that until you understand the basic concept. Now let's do the problems.
.
First problem ... given -x + 4y = 4
.
Find the y-intercept by setting x equal to zero and you get 4y = 4. Solve for y by dividing
both sides by 4 to get y = 1. So the graph crosses the y-axis at +1.
.
Next, find the x-intercept by setting y = 0 and you get -x = 4 which becomes x = -4. So the
graph crosses the x-axis at -4.
.
You can graph this line by plotting the x-intercept point on the x-axis and the y-intercept
point on the y-axis and the drawing a straight line running through these two points.
The graph of -x + 4y = 4 shows these values with the graphed line running through them:
.

.
Second problem. Given: y = -4x -4
.
Set x equal to zero and the function becomes y = -4. So the y-intercept is -4.
.
Then set y = 0 and you get 0 = -4x - 4. Add 4 to both sides and you have 4 = -4x.
Divide both sides by -4 and you have -1 = x. So the x-intercept is at -1.
.
The graph is:
.

.
and it shows the intercepts we found.
.
Hope this helps you to understand the problems and how to solve them.
.

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Laurie ---
.
You probably can just about do these problems in your head once you understand what is going on.
.
Begin by thinking about the coordinate system of axes. A point on the y-axis must have a corresponding
x value of zero. For example, what is the coordinate pair that corresponds to the point +5
on the y-axis. The pair is (0, +5). That means that for a given linear function if you set x
equal to zero, y will equal the intercept value on the y-axis.
.
Similarly, any point on the x-axis will have zero for its y value. Example, the point (-3, 0)
is on the x-axis. So if you set y equal to zero in the given function, the corresponding
value of x will be on the x-axis.
.
Think about that until you understand the basic concept. Now let's do the problems.
.
Given: x - y = 2
.
If x is zero and you solve for y you get -y = 2 which becomes y = -2. So the point (0, -2) is
the y-intercept. Then if y is zero the function becomes x = +2. So the point (2, 0) on the x-axis
is the x-intercept.
.
You can get the graph for each of these three problems by plotting the intercepts
on the x and y axes and drawing a straight line extending through them. The graph for the
first function ... x - y = 2 ... is:
.

.
and this shows the intercept points that we found.
.
Next problem ... given -x + 4y = 4
.
Find the y-intercept by setting x equal to zero and you get 4y = 4. Solve for y by dividing
both sides by 4 to get y = 1. So the graph crosses the y-axis at +1.
.
Next, find the x-intercept by setting y = 0 and you get -x = 4 which becomes x = -4. So the
graph crosses the x-axis at -4. The graph of -x + 4y = 4 shows these values with the graphed
line running through them:
.

.
Final problem. Given: y = -4x -4
.
Set x equal to zero and the function becomes y = -4. So the y-intercept is -4.
.
Then set y = 0 and you get 0 = -4x - 4. Add 4 to both sides and you have 4 = -4x.
Divide both sides by -4 and you have -1 = x. So the x-intercept is at -1.
.
The graph is:
.

.
and it shows the intercepts we found.
.
Hope this helps you to understand the problems and how to solve them.
.
Bucky

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The algebraic rule is that if you multiply two numbers together and they both have the same sign,
the answer is positive.
.
If you multiply two numbers together and they have different signs (one number is positive and
the other is negative) the answer is negative.
.
In this problem, both numbers have the same sign ... both are negative. Therefore, the answer
must be positive. Multiply the 5 times the ll and get 55 and make it a positive answer or +55.
.
Hope this helps you to understand algebraic multiplication a little better.
.

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Start with the equation for the area of a rectangle ...
.
Area = width*length
.
or for short:
.
A = W * L
.
You are told that the area is 24 square meters, so substitute 24 for A to get:
.
24 = W * L
.
Next you are told that the length L is equal to the Width plus 2 meters. In equation form
this is:
.
L = W + 2
.
Return to the area equation and substitute W + 2 for L and you have:
.
24 = W * (W + 2)
.
Note that this equation has only 1 variable ... W. Since it only has one variable,
we have a chance of solving it. Begin by multiplying out the right side:
.
24 = W^2 + 2W
.
Get this into standard quadratic form by subtracting 24 from both sides to get:
.
0 = W^2 + 2W - 24
.
Transpose (switch) the sides of the equation to get it into the more familiar form:
.
W^2 + 2W - 24 = 0
.
Factor the left side:
.
(W + 6)*(W - 4) = 0
.
This equation will be true if one of the factors equals zero because a multiplication
by zero on the left side will make the left side equal zero and therefore equal the zero on the
right side. So set each factor equal to zero and solve for the value of W.
.
W + 6 = 0
.
Subtract 6 from both sides and you get W = -6. But this doesn't make sense ... a negative width???
.
Set the other factor equal to zero:
.
W - 4 = 0
.
Add 4 to both sides and you get W = 4. That makes sense. The width is 4 meters and since
the length is 2 meters longer than the width, the length is 6 meters. If you multiply the
width (4) times the length (6) you get 24 square meters, so that checks.
.
The answers are length = 6 meters and width = 4 meters.
.
Hope this helps you to understand the problem.
.

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Let R represent Renato's present age. Let D represent Doming's present age.
.
Since Renato is 3 years younger that Doming, this means that if we add 3 years to Renato's
age, the answer will equal Doming's age. In equation form this is:
.
R + 3 = D
.
6 years ago, how old was Renato? The answer to that is R - 6. And 6 years ago, how old was
Doming? The answer to that is D - 6. The problem tells you that at that time Renato's age
was two-thirds of Doming's age. So we can say:
.
R - 6 = (2/3)*(D - 6)
.
But we had an earlier equation that said R + 3 equals D. So in this equation we can substitute
R + 3 for D and get:
.
R - 6 = (2/3)*(R + 3 - 6)
.
Inside the second set of parentheses we can combine the +3 and the -6 to get -3. This makes
the equation become:
.
R - 6 = (2/3)*(R - 3)
.
Multiply out the right side and get:
.
R - 6 = (2/3)R - 2
.
You can get rid of the fraction by multiplying the entire equation (both sides and all terms) by
3 to get:
.
3R - 18 = 2R - 6
.
Get rid of the -18 on the left side by adding 18 to both sides and you have:
.
3R = 2R + 12
.
Get rid of the 2R on the right side by subtracting 2R from both sides and you have:
.
R = 12
.
So we can tell that Renato's present age is 12. And since Renato is 3 years younger than
Doming, that means that Doming is 15 years old.
.
Six years ago Renato was 12 - 6 or 6 years old. And six years ago Doming was 15 - 6 = 9 years old.
Renato's age at that time (6) was two-thirds of Doming's age (9), so that checks also.
The answers check ... Renato is presently 12 and Doming is presently 15.
.
Hope this helps you to understand the problem.
.

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The basic idea we will use in solving this problem is the formula that says: Distance equals
speed times time. [You can get a feel for this equation if you think, "If I go at 60 miles per
hour for 2 hours I should be 120 miles (60 * 2) from where I started.] Anyhow, we can write
this distance equation as:
.
D = S * T
.
where D represents distance, S represents speed, and T represents time.
.
The first sentence of the problem tells you that you are going to travel a distance of
10 miles upstream. Traveling upstream means that you are going against the direction that
the stream is flowing, and the actual speed of the boat is its speed in still water (call it
S) minus the speed of the current (call that s and the problem says it is 2 miles per hour).
So when you are going upstream for 10 miles at a speed of (S - 2) miles per hour our distance
equation is:
.

.
Solve this equation for T by dividing both sides of this equation by (S - 2) to get:
.

.
This T is the time it takes to go upstream for 10 miles.
.
Now let's look at the return trip. Since you are going to return to the starting point, you will
again go 10 miles. But this time you will be going in the same direction as the current is
flowing. Therefore, in this direction the speed of the boat will be its speed in still water
PLUS the speed of the current. Therefore, the actual speed of the boat is given by (S + 2)
when you go downstream. And the distance equation for this part of the trip is:
.

.
Solve this equation for the time it takes to go downstream by dividing both sides of the
equation by (S + 2) to get:
.

.
Finally, the problem tells you that the total time of the trip is 2 hours. That means that the
time going upstream ... for which we have:
.

.
plus the time to go downstream ... for which we have:
.

.
must add together to give 2 hours. In equation form this is:
.

.
You can get rid of the denominators by multiplying this entire equation (all terms on
both sides by (S - 2)* (S + 2). When you do that the equation becomes:
.

.
Cancel the terms in the denominators with their corresponding like terms in the numerators as
shown:
.

.
You are then left with:
.

.
Notice that you can simplify this a little by dividing both sides (all terms) by 2 to get:
.

.
Now do the multiplications. First, the left side multiplies out to give:
.

.
Now multiply out the two terms on the right side:
.

.
Combine terms on the right side. Note the +10 and -10 cancel out and you are left with the
equation:
.

.
Get this into standard quadratic form by subtracting 10S from both sides to get:
.

.
The statement in the problem that tells you to round off to the nearest tenth suggests that
this problem does not have an integer solution. So let's skip trying to factor the equation
and go directly to using the quadratic formula. The quadratic formula says that for an equation
of the standard form:
.

.
The solutions will be given by the equation:
.

.
By comparing the equation we are trying to solve with the standard form we can see that
a = 1, b = -10, and c = -4. Plug these values into the appropriate places in the solutions equation
and you have:
.

.
The terms inside the radical become . Substitute that and the equation
becomes:
.

.
Note that the denominator simplifies to 2 and the -(-10) is +10. Substituting these values
results in the further simplified form:
.

.
Calculator time. The square root of 116 is 10.77032961. Substitute that value and you have:
.

.
Look carefully at the numerator. If you use the minus sign the resulting numerator will
be a negative value ... and it doesn't make sense to have a negative speed. So let's just
go with the positive sign and this makes the equation:
.

.
This means that the speed of the boat in still water is (rounding to the nearest tenth) 10.4
miles per hour.
.
Check ... in going upstream the time is equal to distance divided by speed and that is:
.
hours.
.
in going downstream the time is:
.
hours.
.
So the total round trip time is 1.19 + 0.81 = 2 hours ... just as it should be. The answer
checks so we can say that the speed of the boat in still water is 10.4 miles per hour.
.
Hope this helps you understand the problem and how you can get the answer.
.

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Given:
.
5(2a + b) - (a + 3b)
.
Do the distributed multiplication by multiplying 5 times each of the two terms in the first
set of parentheses. When you do you get:
.
10a + 5b - (a + 3b)
.
The second set of parentheses is preceded by a minus sign. Because of the minus sign if you
remove the parentheses you need to change the signs of the terms inside the parentheses.
When you remove these parentheses the expression becomes:
.
10a + 5b - a - 3b
.
Now, combine the like terms. 10a combines with -a to give 9a. And 5b - 3b combines to
give +2b. As a result, the entire expression is simplified to:
.
9a + 2b
.
Hope this helps you to understand the problem ... and some techniques for simplifying
algebraic expressions.
.

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Given:
.

.
Your work was:
.
<=== ok
.
<=== ok
.
<=== mistake in last term
.
You have a mistake in evaluating . Replace 81 with and the term
becomes:
.

.
Bring the exponent 4 out as a multiplier and the result is:
.

.
But
.
so this reduces to:
.

.
So in your last equation replace - 9/4 with -1 and you have:
.

.
and in the second term note that so the expression reduces to:
.

.
That should be enough to convince your teacher that you know what you are doing. I'm not sure
what else could be productively done on this problem. If you feel that it needs more work,
post it again with the above change to your work and see if some other tutor has another idea
about how to go further.
.
Hope this helps you and the study group. Check my work and ensure that I haven't made a "late night"
error.
.

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To solve by addition, one of the equations must have a term that is equal to the same term in
the other equation, but these terms have opposite signs.
.
You are given the two equations:
.
+5x-3y=+13
+4x-3y=+11
.
Notice that the term 3y appears in both equations, but they have the same sign. How about
if we multiply the bottom equation (all terms on both sides) by -1. When we do that the
pair of equations becomes:
.
+5x-3y=+13
-4x+3y=-11
.
Now you can add the two equations in vertical columns. When you do the -3y and the +3y
cancel each other out and the addition results in:
.
x = 2
.
Substitute that value of x into either of the original problems and solve for y. Let's make
that substitution into the first equation. Start with:
.
+5x - 3y = 13
.
Substitute -2 for x to get:
.
5(2) - 3y = 13
.
Multiply out the first term:
.
10 - 3y = 13
.
subtract 10 from both sides of this equation:
.
-3y = +3
.
Divide both sides by -3 to solve for y and you get:
.
y = 3/-3 = -1
.
In summary the answer to this problem is x = 2 and y = -1.
.
Hope this helps you to understand the problem and how to solve it.
.

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One equation you can take advantage of is the form:
.

.
This is called the slope-intercept form and it is the equation of a line in which m, the multiplier
of the x term, is the slope of the line and b is the value on the y-axis where the line crosses the y-axis.
.
In the first problem you correctly found that the slope was -1/5. You can substitute this
value into the slope-intercept form and you get that the equation for the line then becomes:
.

.
At this point you just need to find b to complete the equation. To do this you can use
either of the two points that you were given. Let's use the point (3, 4). This tells you
that when x = 3 and y = 4 the equation must balance. So substitute 3 for x and 4 for y and
the equation becomes:
.

.
Multiply out the first time on the right side to get:
.

.
Solve for b by adding +3/5 to both sides. [This gets rid of the on the right
side.] On the left side when you add you get . This results in
b = . So we can substitute this value for b into the slope-intercept form to get:
.

.
Let's check it out by finding if (-2,5) is a point on the line. Substitute into the slope-intercept
form -2 for x and 5 for y and see if the equation balances.
.

.
Replace 5 by its equivalent and this becomes:
.

.
Multiply out the first term on the right side:
.

.
The two terms on the right side add to and so the equation is true. This checks
the equation
.
The second problem asks you to find the slope and y-intercept of the equation:
.

.
Let's work this into the form
.
Begin by subtracting 4x from both sides to get rid of the 4x on the left side. This changes
the equation to:
.

.
Make the term on the left side positive by multiplying both sides (all terms) by -1 to get:
.

.
Solve for y by dividing both sides (all terms) by 5 to get:
.

.
Note that this is in the slope-intercept form. So just by looking at it you can tell that
the slope is [you can tell that because it is the multiplier of x which is the slope.]
.
And you can tell that the y-intercept is because that is the constant term
in this slope-intercept equation.
.
Hope this helps you to understand these problems. Since it's late, please check my math to
make sure I didn't make a dumb error in the numbers. The basic process of using the slope-
intercept form is correct.
.

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Good Job ...
.
x^2+y^2=25 and x-2y=-5 (changed to x=2y-5) and substituted in to 1st equation
(2y-5)^2+y^2=25 <=== OK
.
4y^2-20y+25+y^2=25 <=== OK
.
(combine like terms)
5y^2-20y+25=25 <=== OK
.
(subtract 25 from each side)
5y^2-20y=0 <====
.
Now just factor 5y out of the two terms on the left side to get:
.
5y*(y - 4) = 0
.
This equation will be true if either of the two factors on the left side equals zero because
a multiplication by zero on the left side makes the left side zero and, therefore,
equal to the zero on the right side.
.
So there are two possible answers for y (because the line crosses the circle at two points).
.
Set the factors equal to zero ...
.
First:
.
5y = 0 ... divide both sides by 5 to get y = 0
.
Next:
.
y - 4 = 0 ... add 4 to both sides ...
.
y = 4
.
Plug these two values for y (0 and 4) into the equation x - 2y = -5. [Of the two equations this will
probably be the easier one to work with.]
.
When y = 0 the equation x - 2y = -5 becomes x = -5. So (-5, 0) is one of the intersection
points.
.
When y = 4 the equation x - 2y = -5 becomes x - (2*4) = -5 which simplifies to x - 8 = -5
.
Add 8 to both sides and you get x = +3. So the point (3, 4) is the second point at which the
line crosses the circle.
.
You did a very good job on the hardest part of the problem ... you just needed a little hint
at the very last part. Keep up the good work ... good luck to you and your study group.
.

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Given:
.

.
If you factor all these polynomials, you get:
.

.
Cancel terms in one of the denominators with the same term in one of the numerators
as follows:
.

.
You are left with the answer:
.

.
Hope this helps you to find where you are having trouble.
.

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Let's begin this problem by getting it into the standard quadratic form:
.

.
You can get it into the quadratic form by adding 1 to both sides of the equation to transform it
to:
.

.
By comparing this to the standard form above you can see that "a" corresponds to 2,
b corresponds to 10, and c corresponds to 1.
.
If you have trouble factoring the original problem or you suspect that the problem may have a
set of complex answers, a good method to use on a quadratic equation is the quadratic formula
which says that for the form:
.

.
the solution is given by:
.

.
for this problem, if you substitute +2 for a, +10 for b, and +1 for c, you get the solution
as:
.

.
Inside the radical sign the simplifies to . Since this is
positive you know that the two values of x will be real. The square root of 92 is 9.591663047.
Substituting this into the equation for x results in:
.

.
The -(10) is equivalent to -10 and in the denominator the 2*2 = 4. These two simplify the
equation for x to:
.

.
This means the two answers for x are:
.

.
and
.

.
Hope this helps you to do the problem. If you want to get the answers in terms of radicals,
you can replace by and your answers will be:
.

.
Which you can simplify down to:
.

.

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This is how I interpret what the problem is asking you to find.
.
When empty, the tank can hold a maximum of 6 cubic feet of water. You find that volume by
multiplying the three dimensions of the tank ... 1 ft times 2 ft times 3 ft = 6 cu ft.
.
When the tank is empty you put the metal cube in it. The question then is how many cubic feet
of water can you then put into the tank before the tank is full.
.
Since the metal cube is 6 inches on a side, each side is foot long. Therefore, the volume
of this cube is the product of or 1/8 cubic ft
.
Once the cube is in the tank, water cannot occupy that 1/8 cubic ft of space. So the amount
of remaining space that water can occupy is the original 6 cubic feet less 1/8 cubic ft
which is 5 7/8 cubic feet or in decimal form 5.875 cubic feet.
.
Hope this helps you to understand the problem and how you can work it.
.

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Percent is defined as "hundredths". So, if you can get any fraction in the form where it
has 100 as its denominator, you can say that the numerator is the percent.
.
In this case you are given:
.
as the fraction.
.
Since the denominator is 100, 29 is the percent. So the answer is 29%
.
Suppose though that you were given:
.

.
and asked what percent that was. To convert the 25 so that it becomes 100 you multiply
it by 4. But if you multiply it by 4 you must also multiply the numerator 3 by 4. When
you do those two multiplications you get:
.

.
Now that you have the denominator equal to 100, you can say the percent is the corresponding
numerator of 12. This means that is equivalent to and this corresponds
to 12%
.
Hope that helps you understand one way to convert a fraction to percent.
.

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Sometimes it's easier to just do a little analysis and not get lost in math. Let's try that
approach.
.
First, the right side of the equation is a quadratic form (the highest power of x is the 2nd
power). From that you know that the graph will be a parabola. And since the sign on the
x-squared term is positive, you can tell that as you move from the left to the right along
the x-axis the graph will drop more and more until it reaches some low point, and then it will
start to rise more and more as you move further to the right. You also know that if you draw
a vertical line through that low point on the graph the graph to the left of that line will
look like the mirror image of the graph to the right of that line.
.
[For your information in other problems of this type, if the sign on the x-squared term
is negative, you can tell that the graph will rise to a peak and then fall back down
the further you go to the right along the x-axis.]
.
Now you can just calculate some points to help you understand what the graph looks like.
.
One nice and easy point on the graph can be found by setting x equal to zero and finding what
the corresponding value of y is. When you set x equal to zero in the given equation for
y, you get:
.
y = (0)^2 - 4 = -4
.
Now you know that the point (0, -4) is on the graph. Plot it.
.
Next let's try letting x = +1 and calculate y. Substituting x into the given equation
results in:
.
y = (1)^2 - 4 = 1 - 4 = -3
.
This time you know that the point (1, -3) is on the graph
.
Next let's set x equal to -1 and calculate the corresponding value of y.
.
y = (-1)^2 - 4 = 1 - 4 = -3
.
Now you know that (-1, -3) is on the graph.
.
Now let's set x = 2 to find that:
.
y = (2)^2 - 4 = 4 - 4 = 0
.
This time you know that (2, 0) is on the graph. Then if you set x = -2 you get the corresponding
value of y to be:
.
y = (-2)^2 - 4 = +4 - 4 = 0
.
The point (-2, 0) is on the graph.
.
If you have plotted these five points on the graph you should have a pretty good idea of what
the parabola looks like. You can use more values for x to find corresponding values of
y and get more points on the graph. With more points you should find that it looks like:
.

.
If you are to "solve" this equation that means that you are looking for the value of x
where the graph crosses the x-axis. If you've done your graphing pretty accurately
you should see that the graph crosses the x-axis at two places ... where x = -2 and x = +2.
This means that for those two values of x the corresponding value of y is zero.
.
Hope this helps you with a little different slant on this problem.
.

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What you have done is listed the integers as being two numbers apart ... consecutive
even integers (such as 10, 12, 14, ...) are two numbers apart. So are consecutive odd integers
(such as 17, 19, 21 ...). But consecutive integers ... such as 11, 12, and 13 ... are just one
apart. Therefore, you represent three consecutive integers as:
.
1st integer: n
2nd integer: n + 1
3rd integer: n + 2
.
The sum of the first and second is n + n + 1 = 2n + 1 and four times that is:
.
4(2n + 1)
.
Seven times the third is 7(n + 2)
.
The problem says that 4(2n + 1) is 17 more than 7(n + 2). So if you take 17 away from 4(2n + 1)
it will equal 7(n + 2). In equation form this is:
.
4(2n + 1) - 17 = 7(n + 2)
.
Multiply out the two distributed multiplications. First multiply 4 times each of the two terms
in its related parentheses. Then multiply 7 times each of the two terms in its related
parentheses. When you do those multiplications the equation becomes:
.
8n + 4 - 17 = 7n + 14
.
Combine the +4 and -17 on the left side and you have:
.
8n - 13 = 7n + 14
.
Get rid of the -13 on the left side by adding 13 to both sides to reduce the equation to:
.
8n = 7n + 27
.
Then get rid of the 7n on the right side by subtracting 7n from both sides. When you do that
subtraction you get:
.
n = 27
.
If n is 27, then the next two consecutive integers are 28 and 29. So the three consecutive
numbers that you are looking for are 27, 28, and 29.
.
Check: 4 times (27 + 28) = 4(55) = 220. 7 times 29 = 203
Is the difference between 220 and 203 equal to 17. Yes it is, so the answer checks.
.
Hope this helps you to see where you made your mistake and what you can do to solve the problem.
.

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Percent is defined as "hundredths" ...
.
So in fraction form 15% is converted to
.
But this fraction can be reduced to a simpler form because both the numerator and denominator
can be divided by the common factor of 5. When you divide the numerator 15 by 5 it becomes 3
and when you divide the denominator 100 by 5 it becomes 20. So the reduced fraction is
and you can't reduce that fraction any further because 3 and 20 cannot be divided
by any number that is common to both of them (other than 1 ... which does not change the problem).
.
So the answer is 15% =
.
Hope this helps ...

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Sort of complex. For reference I'm going to use Roman Numerals to Identify the 4 equations as follows:
.
(I) 2a+3b-4c+6d=6
.
(II) 7a-5b-c=-7
.
(III) 13a-9b=6
.
(IV) d^2-2d=-1
.
Cramer's rule might be an interesting way to solve this, but I'm going to assume that you
haven't studied that yet. So let's just plod our way through.
.
The first thing that I noticed was that equation (IV) has just one variable. So we can solve
it for d first thing. Add 1 to both sides of equation (IV) and it becomes:
.
d^2 - 2d + 1 = 0
.
The left side is a perfect square as follows:
.
(d - 1)^2 = 0
.
To make the left side equal to the zero on the right side we need to have:
.
d - 1 = 0
.
Solve for d by adding 1 to both sides to get:
.
d = 1
.
One down ... only a, b, and c to find. Before we go further let's go back to equation (I) and
substitute 1 for d to make that equation:
.
2a + 3b - 4c + 6(1) = 6
.
subtract 6 from both sides and equation I reduces to:
.
(I) 2a + 3b - 4c = 0
.
Notice that the form of this equation is now very similar to equation (II). Let's multiply
both sides of equation (II) by -4 and it becomes:
.
(II) -28a + 20b + 4c = 28
.
We now have the new equations (I) and (II) as:
.
( I ) 2a + 3b - 4c = 0
(II)-28a +20b + 4c = 28
.
Notice now what happens if we add these two equations vertically in columns. We eliminate
the terms containing the variable c and the equation resulting from this addition (call
it equation (I&II) is:
.
(I&II) -26a + 23b = 28
.
This looks a lot like equation (III) and the pair of equations is now:
.
(I&II)-26a + 23b = 28
(III) +13a - 9b = 6
.
Let's work to eliminate the variable "a". Multiply equation (III) ... all terms on both
sides by 2 to make the equation pair become:
.
(I&II)-26a + 23b = 28
(III) +26a - 18b = 12
.
If we then add the two equations vertically in columns, the two "a" terms cancel each other
and the combined equation that results from the addition (call it equation (I&II&III) becomes:
.
(I&II&III) 5b = 40
.
Divide both sides by 5 and get:
.
b = 40/5 = 8
.
Two down ... only a and c left to find.
.
Now that we know b = 8 we can return to the original equation (III) and substitute
8 for b to get:
.
(III) 13a - 9(8) = 6
.
Multiply 9 times 8 and the equation changes to:
.
(III) 13a - 72 = 6
.
Add 72 to both sides and it further becomes:
.
(III) 13a = 78
.
Solve for a by dividing both sides by 13 to get:
.
a = 78/13 = 6
.
One more to go ... just c left.
.
Return to one of the original equations that contains c and substitute the known values for
a and b. Let's go way back to equation (II). If we substitute 6 for a and 8 for b that equation
becomes:
.
(II) 7(6)- 5(8) - c = -7
.
Do the multiplications and we get:
.
(II) 42 - 40 - c = -7
.
Combine the numbers on the left side:
.
(II) 2 - c = -7
.
Subtract 2 from both sides to reduce this to:
.
(II) - c = -9
.
Solve for +c by multiplying both sides by -1 and we have:
.
c = +9
.
That's it ... no more variables to find. In summary: a = 6, b = 8, c = 9, and d = 1.
.
You can check these answers by returning to the original 4 equations and substituting
the values for a, b, c, and d as needed in each equation. You should (and will) find that
with these values the left side of each equation will equal the right side.
.
Hope this helps you to understand the problem.
.

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To solve equations such as this your goal is to collect the numbers on one side of the equal
sign and the variable on the other side.
.
In this problem you have a -25 that is on the same side as the variable. You need to get rid
of it so that you have just the variable on the left side. You can cancel out the -25 by adding
+25 to the left side. But whatever you do to one side of an equation you must also do to the
other side to keep the equation balanced. So, if you add +25 to the left side, you must also
add +25 to the right side. Let's do it:
.
m - 25 + 25 = -10 + 25
.
On the left side the -25 and the +25 combine to give zero, and there is no need to show that
zero. So the left side becomes just m. On the right side the -10 and the +25 combine to
give +15. Putting these results into the equation gives you the answer:
.
m = +15
.
You have solved for m and you can check this by going back to the original problem of:
.
m - 25 = -10
.
and substituting +15 for m to get:
.
+15 - 25 = -10
.
Notice that on the left side the +15 and the -25 combine to give -10 and that result is
equal to the right side. So the problem checks out if m equals +15. The answer of
m = +15 is correct.
.
Hope this helps you to understand the process of using the equal sign as a sign about which
you collect and simplify the variable terms on one side and also collect and simplify
the number terms on the other side as part of the process to find the answer.
.

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This problem is of a type called distributed multiplication. It tells you to multiply each
of the terms in the set of parentheses by -10
.
When you multiply -10 times 2a you multiply the two numbers together and multiply that by
the variable a. So to get -10 times 2a you multiply -10 times 2 to get -20 and you multiply that
by "a". So the first multiplication in this problem is -10*2a = -20a.
.
Next you multiply -10 times -6b following the same process ... multiply the two numbers and
multiply that result times b. The -10 times the -6 equals +60 (remember that when you multiply
two numbers that both have the same sign, the answer is positive). Then you multiply that
result by b to get the answer of +60b.
.
Putting these two multiplications together results in:
.
-10(2a - 6b) = -20a + 60b
.
Hope this helps you to see how to do a distributed multiplication.
.

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In this problem the variable is the unknown that is represented by the letter y. So you can
say that the variable is y.
.
The coefficient is the number that multiplies the variable. In this case 13 is the number that
multiplies the variable ... so the coefficient is 13.
.
Learn these definitions. They are important. You need to know what they mean so you can communicate
with mathematicians or other students using terms that you both understand.
.
Hope this helps.
.

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This problem is typical of the last step in solving many equations, so you really need to
understand how to do it and why.
.
You are given:
.

.
What you do is to divide both sides of this equation by the number that multiplies the unknown.
In this case, y, the unknown, is multiplied by 16 so you divide both sides by 16 as follows:
.

.
On the left side the 16 in the denominator goes into the 16 in the numerator 1 time. So
on the left side you are left with 1 times y as follows:
.

.
This means that left side is reduced to just y and therefore the equation is simplified to:
.

.
And on the right side the denominator 16 goes into the numerator 64 four times. After this
division you have reduced the equation to:
.

.
And that is the answer to the problem, y equals 4.
.
After you are familiar with problems like this you can work this problem in your head by
telling yourself, "Divide both sides by 16. The left side becomes y and the right side
is 64 divided by 16 which is 4 ... so y equals 4."
.
Hope this helps you to understand this important process that you follow for common problems
of this type.
.

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First problem:
.
You are given the equation of the height of an object that is thrown upward as:
.

.
and the first question you are asked is to find the time that the object reaches its maximum
height. The object starts out at ground level, rises to a peak, and then falls back to
ground level. Neglecting air resistance and other minor considerations, it spends half its
time rising and half its time falling back to ground level. So one way you can find the
time it takes to reach its peak height is to find the time of launch and the total time that
goes by until it hits the ground ... then divide that time by 2.
.
Notice that at ground level the height given by h(t) is zero. So let's substitute zero
into the equation for h(t) and get (after reversing the sides of the equation to get it
into a little more familiar form):
.

.
Note that t is a common factor of both the terms on the left side, so it can be factored to
make the left side become:
.

.
Notice that this equation will be true if either of the factors is equal to zero because a
multiplication by zero on the left side will make the left side equal the zero on the right
side.
.
Setting the first factor [which is t] equal to zero results in:
.

.
This means that at t = 0 seconds the object is at ground launch. No surprise here.
.
Next setting the second factor [which is -16t + 308] equal to zero gives:
.

.
Solve this by first subtracting 308 from both sides to get:
.

.
and then dividing both sides by -16 to get:
.

.
This tells us that from time of launch at t = 0, 19.25 seconds later the object hits the ground.
Since half of that time was spent rising and half of that time was spent falling back down,
the time at which the object reaches its peak is . So at 9.625 seconds
after launch the object is at its maximum height.
.
Another way you can find this time is to apply part of the quadratic formula. Recall that
the quadratic formula applies to quadratic equations of the form:
.

.
If you compare this with your height equation you will see that a = -16, b = 308, and c = 0.
.
Then just use that portion of the quadratic formula that is to find the
time at the peak. Substituting 308 for b and -16 for a results in:
.
seconds.
.
This is the same answer, just a little different way of getting it.
.
Then to find the maximum height, just substitute 9.625 seconds for t in the height equation
to get:
.

.
So the object rises to a height of 1482.25 feet. That's quite a throw!!!! Check your problem
to see if the 308 is the correct multiplier of the t term. If it is, that's really the
answer .... 1482.25 feet up and 1482.25 feet back down again.
.
Next problem.
.
The Area of a square that has S as the length of one side is given by the equation:
.

.
and the Perimeter of the square is the sum of the lengths of all its sides:
.

.
Since the Area is 12 more than the Perimeter, if you take 12 away from the Area, the result
will equal the Perimeter. In equation form this is:
.

.
Get this into the standard quadratic form by subtracting 4S from both sides to get:
.

.
This equation factors to:
.

.
As in the previous problem, this equation will be true if either factor equals zero.
.
Setting the first factor equal to zero results in:
.

.
Subtract 2 from both sides and you get:
.

.
This answer doesn't make sense ... a side of minus 2 length??? Ignore it.
.
Setting the second factor equal to zero gives:
.

.
Add 6 to both sides and you have:
.

.
This looks better. A square with a side of 6 has an area of 6^2 = 36 and a perimeter of
6*4 = 24. The area is 12 more than the perimeter, just as the problem specified. So the
side length you were to find is S = 6.
.
Hope this helps with your understanding of these two problems. It's pretty late so you had
better use your calculator to check my math. The process is correct, but I may have let
a calculation error slip in ... I don't think so but better safe than sorry ...
.

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One step at a time. First, solve the equation for y:
.
Begin by getting rid of the term 6x on the left side so that you just have the term containing
the y alone on the left side. Do this by subtracting 6x from both sides to get:
.

.
You are trying to solve for +y so at this point you may want to change the sign of -5y to +5y.
You can do that by multiplying both sides of the equation (all terms) by -1 to change the
equation to:
.

.
Finally, solve for y by dividing both sides of this equation by 5 ... the multiplier of
y to get:
.

.
Notice that the equation we now have is:
.

.
and this is in the slope-intercept form:
.

.
in which m, the multiplier of x, is the slope of the graph and b is the value on the y-axis
where the graph crosses the y-axis. By comparing your equation with the slope intercept form
you can see that the graph of your equation has a slope of and it crosses the
y-axis at the value of +1 on the y-axis.
.
Now look at the other equation you were given ... namely:
.

.
Comparing this equation to the slope intercept form you will see that it also has a slope
of but its graph crosses the y-axis at .
.
Now recognize that two graphs having the same slope but different crossing points on the
y-axis are parallel lines that are always separated in vertical distance by an amount equal
to the difference on the y-axis equal to the crossing points. The graph of the two equations
shows this. The "red" graph is the graph of the equation and the green
graph is the graph of the equation
.

.
Hope this helps you to understand the problem and shows you that lines given by the two equations
are actually parallel.
.

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Begin by solving the given equation for x. The first step is to get rid of the +6 on the left
side by subtracting 6 from both sides. When you do this subtraction the equation becomes:
.
2x = -6
.
Then solve for x by dividing both sides of this equation by 2 because 2 is the multiplier of x.
When you do this division the equation is reduced to:
.
x = -6/2 = -3
.
The question now is, "How do you graph x = -3?" One way to look at this is that x is always
equal to -3 no matter what value y is because y doesn't even appear in the equation. Therefore,
you can say things such as when y = 0, x = -3, when y = -10, x = -3, when y = +8, x = -3
and so on. If you plot these points ... (-3, 0), (-3, -10), (-3, +8) you will begin to
get the idea that on a coordinate system the graph of x = -3 is a vertical line that goes
through the point -3 on the x-axis.
.
Your graph should look like this (the graph is the vertical line in light red):
.

.
Hope this helps. Be aware that the graph of any equation of the form x = A where A is a constant
is a vertical line through the point A on the x-axis.
.

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You need to know a couple of properties of logarithms.
.
One of the properties is the multiplication property:
.
The first problem is to find . Note that 6 is equal to 2*3. So you can re-write
the problem as But the property of logs described above converts this to:
.

.
Both of these logs are given ... and Inserting
these values into the problem results in:
.

.
So the answer is
.
The next problem uses the exponent property that
.
The problem is to find the value of . To do that you can recognize that
. If you substitute that into the problem, you get:
.

.
Next apply the exponent property to add another step to the problem:
.

.
Finally add the last step by substituting 0.683 for :
.

.
So the answer to this second problem is
.
The final problem is:
.

.
Use the multiplication property to split this into two logarithms as follows:
.

.
Another property is that for any base if you take the logarithm of the base, the answer is 1.
This means . Plus you are given that . Making these
two substitutions results in:
.
.
.
So the answer to this problem is:
.
Hope this helps you to gain some insight into a couple of properties of logarithms. It's
sort of late to be doing math, so be sure to check the math in these problems. Mistakes
have a tendency to creep into late-night work ... but the basic principles involving the
properties of logarithms are correct.
.

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Probably the easiest way to solve this equation is by factoring. Since both terms on the left
side contain an x, you can factor out an x to give:
.
x*(x - 14) = 0
.
This equation will be true if either of the factors on the left side are zero because a
multiplication by zero on the left side will make the left side equal the zero on the right side.
.
So we can get the two answers by setting each of the factors equal to zero.
.
For the first factor equal to zero we have:
.
x = 0 and that is one answer.
.
For the second factor equal to zero we have:
.
x - 14 = 0
.
And if we add 14 to both sides we get:
.
x = 14 and that is the second answer.
.
Check ... go to the original equation x^2 - 14x = 0 and substitute 0 for x. The result is:
.
0 - 0 = 0
.
and that is a correct equation, so the answer x = 0 checks.
.
Return to the original equation x^2 - 14x = 0 and substitute 14 for x. When you do you get:
.
14^2 - 14*14 = 0
.
and since 14^2 = 14*14 the equation becomes:
.
14*14 - 14*14 = 0
.
This also is a correct equation so the answer x = 14 checks.
.
Hope this helps you to understand the problem.
.