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Let H represent the height of the individuals in rungs and t represent the time in seconds that
passes while they climb.
.
Bob starts at rung 30 and goes down at a rate of -2 rungs per second. So the equation for
Bob's height is:
.
H = -2*t + 30
.
Meanwhile, Rob starts at rung 0 and goes up at a rate of +1 rung per second. So the equation
for Rob's height is:
.
H = t + 0 = t
.
You are interested in finding the height (in rungs) where the two will be standing on
the same rung. Since the two heights are the same, we can set the right sides of our two
height equations equal. So we can say that:
.
-2*t + 30 = t
.
To solve this equation for t, first get rid of the 30 on the left side by subtracting
30 from both sides of the equation. This subtraction converts the equation to:
.
-2*t = t - 30
.
Next get rid of the t on the right side by subtracting t from both sides. The equation then
becomes:
.
-3*t = -30
.
Solve for t by dividing both sides of this equation by -3 to find that:
.
t = -30/-3 = 10 seconds
.
In 10 seconds Rob will have climbed to rung number 10 because he climbs at 1 rung per
second. And in 10 seconds at a rate of 2 rungs per second Bob will climb down 20 rungs. But
Bob started on rung 30 so he will be at rung 10 also because 30 - 20 = 10.
.
Hope this helps you to understand the problem a little better.
.

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Call the two angles A and B
.
You are told that the two angles are supplementary. That means that when you add the measure
of angle A (call it mA) and the measure of angle B (call it mB) the resulting sum is 180 degrees.
This relationship can be written in equation form as:
.
mA + mB = 180
.
You are also told that the two angles are congruent. This means that their measures
are equal. You can write this relationship as the equation:
.
mA = mB
.
From this second equation you can see that wherever you have mA you can substitute mB
in its place because they are equals. So go back to the equation:
.
mA + mB = 180
.
In place of mA substitute mB. This makes the equation become:
.
mB + mB = 180
.
On the left side you can see that the sum is 2 times mB or 2*mB. Make this simplification
to get:
.
2*mB = 180
.
To solve for mB divide both sides of this equation by 2. When you do that division the
equation reduces to:
.
mB = 180/2 = 90
.
This tells you that the measure of angle B is 90 degrees, and that means that the measure
of angle A is also equal to 90 degrees because the two angles are congruent.
.
Hope this helps you to understand the problem.
.

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All of these equations are in the slope-intercept form. This form is:
.
y = m*x + b
.
where m ... the multiplier of x ... is the slope and b is the value on the y-axis where the
graph crosses the y-axis.
.
From this you can tell that the slope of the graph for equation (a) is (5/3) because (5/3)
is the multiplier of x.
.
Similarly the slope of (b) is (3/5)
.
The slope of (c) is 2
.
and the slope of (d) is 2
.
If the lines are to be parallel, they must have the same slopes. (You might be able to
visualize that if the slopes are different, the lines have to intersect at some place.)
.
Notice that equations (c) and (d) both have the same slope and therefore, they are the
pair of graphs that are parallel.
.
To help you visualize the graphs ... here they are. The graph of (a) is in "red" ... (b) is
in green ... (c) is in blue ... and (d) is in purple. You can see that the blue and purple
graphs (c) and (d) are parallel.
.

.
Hope this helps you to understand the problem.
.

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The list price is the original price ... that is the price the article was before a 12% discount
was taken.
.
The list price minus the 12% of the list price mark-down results in the $4400 discounted
price. If L represents the list price the equation for the discount is:
.
L - 0.12L = 4400
.
which represents the list price minus 12 percent (0.12) of the list price resulting in
4400 dollars.
.
Doing the subtraction on the left side reduces the equation to:
.
0.88L = 4400
.
You can now solve for L by dividing both sides of the equation by 0.88 to get:
.
L = 4400/0.88 = 5000
.
So the original list price was $5000.00
.
Check:
.
The original list price was $5000. The 12% discount was 0.12 times $5000 and this product
is $600. So the discounted price is $5000 less $600 which does equal $4400, just as the problem
said it should.
.
Hope this helps you to understand the problem a little better.
.

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Think about a coordinate system. If a point is on the y-axis, what is its corresponding
x-value. You can see that any point on the y-axis has zero for its x-value.
.
Therefore for the given equation, if you set the value of x at zero, the resulting value of y
will be the value on the y-axis where the intercept occurs. So ... start with:
.
-x + 3y = 15
.
Set x equal to zero and the equation reduces to:
.
3y = 15
.
Solve for y by dividing both sides of this equation by 3 to get:
.
y = 15/3 = 5
.
The graph crosses the y-axis where the value of y is +5. You may also see the y intercept
given in the form of (x, y). So, from the discussion above, in this form the y-intercept
is expressed as (0, +5).
.
The graph of the given equation looks like this:
.

.
and you can see the y-intercept on this graph is at y = +5
.
Hope this helps you to understand the problem a little better.
.

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The formula for converting a Fahrenheit temperature to a Celsius temperature is:
.

.
If you substitute 104 for F, the equation becomes:
.

.
Inside the parentheses you can subtract 32 from 104 and you get 72. This reduces the equation to:
.

.
Divide the 9 of the denominator into the 72 of the numerator to get 8. The equation then
becomes:
.

.
This tells you that 40 degrees Celsius is equivalent to 104 degrees Fahrenheit.
.
Hope this helps you to understand the process for converting Fahrenheit temperatures to
their corresponding Celsius temperature.
.

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Given:
.

.
You can convert the denominator of this fraction to a real number by multiplying the denominator
by its conjugate ... the same term as the denominator only with the opposite sign between the
terms. In this case, the conjugate is . If you multiply the denominator by its
conjugate, you must also multiply the numerator by the same conjugate.
.
This multiplication leads to:
.

.
When you multiply the denominators:
.

.
You can do so by multiplying the 4 in the first set of parentheses by both terms in the second
set of parentheses and then multiplying the -7i from the first set of parentheses by
both terms in the second set of parentheses:
.

.
Notice that the +28i and the -28i are equal but of opposite sign. Therefore, they cancel
each other out and you are left with:
.

.
But, by definition, . Substitute -1 for and the expression becomes:
.

.
So the denominator, when multiplied by its conjugate, becomes 65.
.
The numerator, when multiplied by the conjugate, is:
.

.
This numerator is then over the denominator 65 to give:
.

.
So the answer to this problem, in the form a + bi, is:
.

.
Hope this helps you to understand the problem.
.

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Given: the parabola
.

.
To find the intersections with the x-axis, set y = 0 because any point on the x-axis has
zero as its y-value. So the equation to solve then becomes:
.

.
Solve this by getting rid of the -4 on the right side by adding +4 to both sides. On the right
side the -4 and the +4 cancel each other when they are added. So when you add +4 to both sides
the equation becomes:
.

.
Solve for x by taking the square root of both sides to get two answers:
.
and
.
So the intersections of the parabola and the x-axis occur on the x-axis at x = -2 and at
x = +2. These points are P and Q. And, if you subtract them you get:
.

.
The points P and Q are 4 units apart.
.
You can see this in the graph of the parabola shown below:
.

.
Hope this helps you to understand the problem.

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The equation for the circumference of a circle is:
.

.
Where C represents the circumference and R represents the radius.
.
The problem tells you that C = 56.52 inches and . Substituting these two values
into the equation for the circumference results in:
.

.
Multiply out the constants on the right side to get:
.

.
Solve for R by dividing both sides of the equation by 6.28 to get:
.

.
The radius of the circle is 9 inches.
.
The area of a circle is given by the formula:
.

.
in which A represents the area, again , and for this problem R = 9 inches.
Substituting the values into the equation results in:
.

.
The area of the circle is, therefore, 254.34 square inches. The problem tells you to round
this to the nearest tenth of an inch. Rounding gives you the final answer of 254.3 square inches.
.
So the answer is that the Area of this circle is 254.3 square inches.
.
Hope this answer helps you to see your way through this problem.
.

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Let D be the number of dimes and Q be the number of quarters.
.
Since there are 3 times as many quarters as dimes, if you multiply the number of dimes by 3
the answer will be the same as the number of quarters. In equation form this becomes:
.
3D = Q
.
Each dime is 0.1 of a dollar. So if you multiply the number of dimes D by 0.1 you get the
amount in dollars that come from dimes. The resulting term is 0.1D.
.
Each quarter is 0.25 of a dollar. So if you multiply the number of quarters Q by 0.25 you get
the amount in dollars that come from quarters. The resulting term is 0.25Q.
.
The problem tells you that adding these two terms will result in $6.80. In equation
form this is:
.
0.1D + 0.25Q = 6.80
.
From our first equation we know that Q = 3D. This tells us that we can go to our dollar equation
and substitute 3D for Q. This substitution leads to:
.
0.1D + 0.25(3D) = 6.80
.
Multiply out the second term on the left side and you get:
.
0.1D + 0.75D = 6.80
.
Add the two terms on the left side to get:
.
0.85D = 6.80
.
Solve for D by dividing both sides of this equation by 0.85:
.
D = 6.80/0.85 = 8
.
This tells you that the number of dimes equals 8
.
Since the number of quarters is three times the number of dimes, then the number of quarters
is 8*3 = 24.
.
Let's check ... each 4 quarters is a dollar ... and there are six groups of 4 quarters, then
the number of dollars from quarters is $6.00
.
and the 8 dimes is 80 cents or $0.80.
.
The sum of these two amounts is $6.00 + $0.80 = $6.80, just as the problem said it should.
.
And with 24 quarters and 8 dimes, there are 3 times as many quarters as there are dimes.
This is also as the problem said it should be. Therefore, the answer of 24 quarters and
8 dimes fully checks out.
.
Hope this helps you to understand the problem and how to solve it algebraically.
.

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The first sentence of the problem indicates that you are looking for two numbers. It also tells
you that there is a larger number (let's call it L) and as smaller number (let's call it S).
Finally, it tells you that 3 times the larger (3*L) is equal to 4 times the smaller (4*S).
In equation form this can be written as:
.
3*L = 4*S
.
The sum of the numbers is 21. Therefore, another equation is:
.
L + S = 21
.
Let's solve these two equations by substitution. We can start by solving the second equation
for L by subtracting S from both sides to leave just L by itself on the left side. When
you subtract S from both sides, the equation becomes:
.
L = 21 - S
.
Now let's substitute the right side of this equation for L in the first equation that we wrote.
The first equation is:
.
3*L = 4*S
.
substituting 21 - S for L changes the equation to:
.
3*(21 - S) = 4*S
.
Do the distributed multiplication on the left side by multiplying 3 times each of the terms
in the parentheses. 3 times 21 is 63 and 3 times -S is - 3*S. This changes the left side and
the equation becomes:
.
63 - 3*S = 4*S
.
Get rid of the -3*S on the left side by adding +3*S to both sides. On the left side, this
addition cancels the -3*S and on the right side the 3*S and 4*S add to give 7*S. So the
changed equation is:
.
63 = 7*S
.
Solve for S by dividing both sides of the equation by 7. This makes the answer for S:
.
S = 63/7 = 9
.
Now you know that the smaller number is 9. And since the total of the two numbers is
21, the larger number is 21 - 9 = 12.
.
Check: 3 times the larger number is 3*12 = 36. And 4 times the smaller number is 4*9 = 36.
This means that 3*L = 4*S
.
And L + S is 12 + 9 and that does equal 21, just as it should.
.
Therefore, the answers of 12 and 9 satisfy the problem and are, therefore, the correct answers.
.
Hope this helps you to understand the problem.
.

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Call one of the investments x dollars and the other investment y dollars.
.
The way Melinda has her money invested she makes 6% on investment x and 5% on investment y.
This is equivalent to 0.06*x and 0.05*y. These two returns add up to $234.50. So we can
write this in equation form as:
.
0.06*x + 0.05*y = 234.50
.
If the investments were interchanged, then x would return 5% and y would return 6%. So this
time the returns would be written 0.05*x and 0.06*y. If you add these two together, then
the return would be $5.10 more than she got before. This means the return would now be
$234.50 + $5.10 = $239.60. So, in equation form, this time the equation is:
.
0.05*x + 0.06*y = 239.60
.
So we have two equations:
.
0.06*x + 0.05*y = 234.50 and
0.05*x + 0.06*y = 239.60
.
Let's use variable elimination to solve these two equations. Multiply the top equation (all
terms on both sides) by 500 to get:
.
30*x + 25*y = 117250
.
Next multiply the bottom equation by 600 (all terms on both sides) and the equation
becomes:
.
30*x + 36*y = 143760
.
So we have converted the two equations to:
.
30*x + 25*y = 117250 and
30*x + 36*y = 143760
.
Notice now that if we subtract the two equations in vertical columns the term 30*x will
disappear from the resulting equation because we made it such that both equations had
a common x-term. (We did that by our choice of the multipliers for each equation.)
.
Subtracting the two equations vertically, we get:
.
-11*y = -26510
.
Solve for y by dividing both sides of this equation by -11 and you get:
.
y = -26510/-11 = 2410
.
So one of the sums of money (the one she has invested at 5%) is $2410.00
.
We can now return to either of the original equations we developed and solve for x by
substituting $2410 for y. Let's return to the equation:
.
0.06*x + 0.05*y = 234.50
.
and substituting $2410 for y we get:
.
0.06*x + (0.05)(2410) = 234.50
.
Multiply out the second term on the left side and the equation becomes:
.
0.06*x + 120.50 = 234.50
.
Get rid of the 120.50 on the left side by subtracting 120.50 from both sides. When you do
that the equation reduces to:
.
0.06*x = 114
.
Solve for x by dividing both sides by 0.06 and you get:
.
x = 114/0.06 = 1900
.
This tells you that she has $1900.00 invested at 6%.
.
So the answer to this problem is that she has $1900.00 invested at 6% and $2410.00 invested
at 5%.
.
Hope this helps you to understand the problem.
.

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The circumference of the wheel is found using the formula:
.
C = pi*D
.
where C is the circumference and D is the diameter. So for this problem:
.
C = pi*1.5 = 3.14*1.5 = 4.71 feet
.
Each turn of the wheel moves the car forward a distance equal to the circumference of the wheel
.
To move the car forward 37.68 feet the number of rotations (revolutions) the wheel must make
is found by dividing 4.71 into 37.68 ....
.
Rotations = 37.68/4.71 = 8
.
So, if the wheel rotates 8 times, the car will move forward 37.68 feet.
.
Hope this helps you to understand the problem.
.

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You can solve this problem using a proportion. Let L equal the length of the object and
S equal the length of the corresponding shadow.
.
Since the length of the pole is 4-feet and its corresponding shadow is 3 feet, we can set the
ratio of L (the pole length) to S (the corresponding shadow length) and get:
.

.
For the tree, the ratio must be the same way ... that is the unknown length must be the numerator
and the corresponding shadow length of its shadow must be the denominator. So for the tree
the ratio of its length of the shadow length can be written as:
.

.
We can now establish a proportion by setting these two ratios equal to get:
.

.
One way to solve this is to make the denominators on both sides equal. You can do this
by multiplying the numerator and the denominator of the ratio on the left side by 2 to
get:
.

.
This shows you that for these two ratios to be equal, since the denominators are equal, the
numerators must be equal also. Therefore, L must equal 8 which tells you that the tree
is 8 feet tall.
.
Proportions can also be solved by cross multiplying ... multiply each numerator by the denominator
on the opposite side and set the two products equal. For this problem you start with the
proportion:
.

.
Multiply the 4 times the 6 to get 24. Then multiply the L times the 3 to get 3L. Set the
two products equal and you have:
.

.
Solve for L by dividing both sides of this equation by 3 and you have:
.

.
This is the same answer as we got the other way ... the tree is 8 feet tall.
.
Hope this helps you to understand the problem.
.

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Given the sequence (-1, -5, -9, -13, ...)
.
To find the function, first note that the difference between terms is -4 which has to be
added to the preceding term to get the next term in the series.
.
The first term is -1. So you can say that it equals 3 + (-4*1). The second term is then equal
to 3 + (-4*2) and the third term is 3 + (-4*3) and so on.
.
Therefore, to calculate you can use the relationship:
.

.
where n represents the sequence number of the term you are looking for in the series. For example,
to find the 6th term in the series, substitute 6 for n and you have:
.

.
Hope this helps you to understand the problem and shows you a thought process that will
lead you to the answer.
.

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Let C represent the total cost of renting x number of videos.
.
For the video club that charges a $25 fee to join, the cost will be that $25 plus an additional
charge of $2.50 times the number of videos rented (x videos). So the total cost of renting can be
written in equation form as:
.
C = 2.50*x + 25
.
The other video club just charges $3.50 for each video that is rented. So the total cost
to rent x videos is just $3.50 times x. In equation form this is:
.
C = 3.50*x
.
The "break-even" point occurs when the two costs are equal. If the two costs are equal, then
the right sides of the two equations are equal. Set them equal and you have:
.
2.50*x + 25 = 3.50*x
.
You can solve this equation by first collecting the x-terms on one side of the equation. To
do this, get rid of the 2.50*x on the left side by subtracting 2.50*x from both sides of
the equation. When you do that subtraction you end up with:
.
25 = 1.0*x or just 25 = x
.
This tells you that when you rent 25 videos the cost is the same for each club. After that,
the cost will be less for the club with the $25 fee, because each video rental beyond 25 rentals
will cost $2.50 ... but each video beyond 25 in the other club will always cost $3.50 no matter
how many you rent.
.
You can also see this if you graph the two equations. The graph below has the rental cost
on the y-axis and the number of rentals along the x-axis. The "red" graph shows the cost of
renting tapes from the club that has the $25 fee. The green graph shows the cost of renting
tapes from the club that charges $3.50 for each rental.
.

.
Note that out to 25 on the x axis, the green graph is below the red graph. That means the total
cost of getting up to 25 rentals will be lower than the cost of renting from the club with the
$25 fee. For more than 25 rentals (on the x-axis) the total cost of the red graph will
be lower than the total cost of the green graph. Therefore, the cost will be less for the
club with the $25 fee (shown by the red graph being lower than the green graph).
.
Hope this helps you to understand the problem.
.

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Given:
.

.
To solve this equation for x, first get rid of the -7 on the left side. You can do that by
adding +7 to both sides. Adding +7 to the left side cancels the -7 on the left side and
the equation becomes:
.

.
Now solve for x by dividing both sides by 2 and you have:
.

.
This means that x is always 3.5 ... so any point that has 3.5 for its x value will satisfy
the equation you were originally given.
.
As examples, all of the following points would satisfy the original equation:
.
(3.5, 10)
(3.5, 0)
(3.5, -15)
.
because each of those points has an x value of 3.5
.
How would the graph look on a coordinate system?
.
If you plotted the three points above (and any other point that has 3.5 as its value for x)
the points would line up vertically through the point +3.5 on the x-axis, and the graph
would look like:
.

.
Hope this helps you to understand the problem.

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Given:
.

.
Let's start by getting rid of the +4 on the left side. Do that by subtracting 4 from both
sides and you have:
.

.
Next, on the left side of this equation let's group all the terms containing x and all the terms
containing y so that we have:
.

.
Let's complete the square for the contents of each set of parentheses. Do this by taking
half of the coefficient of the first degree term, squaring it, and adding the square in
that set of parentheses. (To counter that you also have to subtract the same amount from
the same side of the equation.)
.
In the set of parentheses that contain the x terms, take half of the -6 to get -3. Square
that to get +9. Add +9 inside the parentheses and to counter that subtract 9 on the same side
of the equation. As a result you get:
.

.
Use the same process for the set of parentheses containing the y terms. Take half of the
-4 to get -2 and then square the -2 to get +4. Put +4 in the parentheses and subtract
-4 elsewhere on the same side. This results in:
.

.
Note that we have created perfect squares in each set of parentheses. Expressing the contents
of the parentheses as squares results in:
.

.
Combine the two constants on the left side and you have:
.

.
Get rid of the constant on the left side by adding 13 to both sides. This results in:
.

.
This is now in the standard form of the equation of a circle. The center of the circle can
be found by setting each of the contents of the parentheses equal to zero as follows:
.
and solve for x by adding +3 to both sides to get
.
then:
.
and solve for y by adding +2 to both sides to get
.
So the center of the circle is (3, 2).
.
The radius of the circle is the square root of the right side of the equation for this
circle. So the radius of this circle is So the radius of the circle is
.
Hope this helps you to understand the problem and how you solve problems involving the
equation of a circle.
.

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You are given that the roots of a quadratic equation are:
.
and
.
and you are asked to find the quadratic equation that has those roots.
.
Since these are roots, you know that they are the values of x that solve a quadratic
expression that is equal to zero. Therefore, you know that:
.

.
is one answer and
.

.
is the other answer.
.
You can convert these two answers to the factors of the quadratic by getting everything
on the right side of these two equalities. So let's start with:
.

.
and subtract from both sides. When you do that subtraction, you get:
.

.
This tells you that (x + 1 - 4i) is one of the factors of the quadratic.
.
Similarly, next go to the other root:
.

.
Convert it to a factor by subtracting from both sides. When you do that subtraction
you get:
.

.
and this tells you that (x + 1 + 4i) is the other factor of the quadratic.
.
To get the quadratic equation, multiply these two factors together and set the product equal
to zero.
.
So you multiply (x + 1 - 4i) times (x + 1 + 4i) and set the result equal to zero.
.
This multiplication requires that you multiply each term in the second set of parentheses
by each term in the first set of parentheses and then combine terms. So let's take the x
from the first set of parentheses and multiply it times everything in the second set of parentheses.
In other words multiply:
.
to get
.
Next take the +1 from the first set of parentheses and multiply it times everything in
the second set of parentheses:
.
to get
.
Finally take the -4i from the first set of parentheses and multiply it times everything in
the second set of parentheses:
.
to get
.
In the third term of this last product, note that by definition and if in
you replace by -1 you get . So the third multiplication
for which you got can be simplified to
.
Now combine all three of the products by adding them:
.

.
Group the common terms and you get:
.

.
And when you combine the grouped terms some of them cancel out. You are left with:
.

.
In the last step you set this result equal to zero and you have as the quadratic equation:
.

.
Hope this helps you to find your error. In this case your teacher is correct.
.

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Downstream is the direction that a river or stream flows. If you toss something that floats
into a stream, it will drift downstream.
.
Upstream is the opposite direction. It means going in the direction that is against the
flow of the water in a stream or river.

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You can view this in a couple of ways.
.
One way is to use the following rules:
.
If a set of parentheses is preceded by a + sign you can just remove the parentheses without
changing the terms inside the parentheses.
.
However, if a set of parentheses is preceded by a - sign you can remove the parentheses
if you also change the signs of the terms inside the parentheses.
.
Or, if the parentheses are preceded by a minus sign, you can view it as a distributed
multiplication by -1.
.
Following the first pair of rules about parentheses for this problem, since the given expression:
.
-(z+92)
.
is preceded by a minus sign, you can remove the parentheses if at the same time you change
the signs of the terms inside the parentheses to get:
.
-z - 92
.
The second way of viewing this problem is to view it as a distributed multiplication
of:
.
-1(z + 92)
.
When you multiply the -1 times each of the terms inside the parentheses, the result is:
.
-1*z + (-1)*92 = -z - 92
.
In either case, the result is the same ... -z - 92
.
Hope one of these ways helps you to understand the problem.
.

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Given:
.

.
First do the distributed multiplication of -5 times each of the terms inside the parentheses.
.
When you multiply -5 times 3x you get -15x. And when you multiply -5 times -3 you get +15.
Substituting these multiplications into the given problem converts it to:
.

.
On the left side combine the constants +6 and +15 to get +21 which reduces the equation to:
.

.
Get rid of the +21 on the left side by adding -21 to both sides. (This is the same as
subtracting 21 from both sides). When you do this addition, on the left side the +21 and -21
combine to cancel out. On the left side the -54 and -21 combine to -75. So the resulting
equation is:
.

.
You can get rid of the minus signs by multiplying both sides of the equation by minus 1 to get:
.

.
Finally, solve for x by dividing both sides of this equation by 15 to get:
.

.
Now that you know that x = 5, you can check the problem by starting with:
.

.
Then substitute 5 for x to get:
.

.
Multiply the 3 times 5 to get:
.

.
Within the parentheses the 15 and -3 combine to 12:
.

.
The -5 times 12 results in -60 and this reduces the equation to:
.

.
The left side terms combine to -54 so the equation further reduces to:
.

.
and this, of course, is true. So our answer checks, and x = 5 is the answer.
.
Hope this helps cure you of your case of "morning brain dead." Press on ...
.

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Given:
.
Body Temperature You may have heard that the average temperature of the human body is 98.6 degrees.
Recent experiments show that the actual figure is closer to 98.2 degrees.† The
figure of
98.6 comes from experiments done by Carl Wunderlich in 1868. But Wunderlich measured the
temperatures in degrees Celsius and rounded the average to the nearest degree, giving
37 degree C as the average temperature.
.
Comment: this is some interesting historical information. It is background for the problem, but
really doesn't affect the mechanics of the rest of the problem. Students that have trouble with
"word problems" often have difficulty because they can't separate such "extra" information from
what they are being asked to do for the problem. So you may want to read the above info for
furthering your background knowledge of history and science, but you don't need it for solving
this problem.
.
The heart of what this problem asks you to do is to convert some Celsius temperatures to
Fahrenheit temperatures. The equation for doing such conversions is:
.

.
where F represents Fahrenheit degrees and C represents Celsius degrees.
.
Now that we have this information, let's work the two problems:
.
a. What is the Fahrenheit equivalent of 37 degree C?
.
All you have to do is to go to the conversion equation and in it substitute 37 for C. So start
with:
.

.
Replace the C with 37 to get:
.

.
Determine the first term on the right side. You can do that by multiplying the 9 times the
37 to get 333. Then divide the 333 by 5 to get 66.6. Substitute this for the first term
on the right and the equation becomes:
.

.
Now just add the two numbers on the right side and you get the sum as 98.6
.
So the answer to this first problem is that 37 degrees Celsius is equal to 98.6 degrees Fahrenheit.
.
Not too bad to do ...
.
Now for the second problem.
.
b. Given that Wunderlich rounded to the nearest degree Celsius, his experiments tell us that
the actual average human body temperature is somewhere between 36.5 degree C
and 37.5 degree C.
Find what this range corresponds to in degrees Fahrenheit.
.
When you discard all the "extra information" in this problem, you are just being asked to
find what the temperature range of 36.5 degrees C to 37.5 degrees C corresponds to in degrees F.
.
So let's return to the equation for converting degrees C to degrees F:
.

.
And first substitute 36.5 for C in the equation to get:
.

.
You can multiply the 9 times 36.5 to get 328.5 and then divide the 328.5 by the denominator 5
and the answer becomes 65.7. Substitute this for the first term on the right side to reduce
the equation to:
.

.
And when you add the two numbers on the right side you get 97.7 degrees as the Fahrenheit
equivalent of 36.5 degrees C.
.
Next you convert 37.5 degrees C to degrees F by using the same method:
.

.
So the Fahrenheit equivalent of 37.5 degrees C is 99.5 degrees F.
.
This tells us that the temperature range of 36.5 degrees C to 37.5 degrees C is (in Fahrenheit)
97.7 degrees to 99.5 degrees.
.
That's what the problem was really asking you to do.
.
Hope this helps you to see your way through it. Just remember the equation for converting
Celsius temperatures to Fahrenheit is: . Just plug in Celsius
temperatures for C, apply the algebra rules of multiplication, division, and addition, and
the answer comes out in Fahrenheit.
.

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The perimeter of a polygon is the sum of the lengths of all its sides.
.
If you add the sides of the given polygon you have:
.
8x + 3 + 6x + 4 + 6x - 3
.
If you combine the terms that contain x you get 8x + 6x + 6x = 20x
.
If you combine the constants you get 3 + 4 - 3 = +4
.
So the sum of all the sides reduces to 20x + 4 and since it is the perimeter, you can set it
equal to 84 as follows:
.
20x + 4 = 84
.
Solve by first getting rid of the 4 on the left side. Do that by subtracting 4 from both
sides of the equation to get:
.
20x = 80
.
Then find x by dividing both sides of this equation by 20 ... the multiplier of the x. This
division leads to:
.
x = 80/20 = 4
.
Now that you know x is equal to 4 you can find the lengths of the three sides by substituting
4 for x in each of the three sides to get:
.
8x + 3 = 8(4) + 3 = 32 + 3 = 35
6x + 4 = 6(4) + 4 = 24 + 4 = 28
6x - 3 = 6(4) - 3 = 24 - 3 = 21
.
So the lengths of the three sides are 35, 28, and 21. If you add these three sides you
find that the perimeter is 35 + 28 + 21 = 84 ... just as the problem said it should be.
This check verifies that the answers are correct.
.
Hope this helps you to understand the problem.
.

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The slope-intercept form of an equation is:
.
y = mx + b
.
in which m, the multiplier of x, is the slope and b, a constant, is the y-intercept.
.
The equation you are given is y = (-2/3)x + 20 and you can see by inspection that this equation
is in the slope-intercept form. By comparing this equation with the slope-intercept
form, you can see that the multiplier of the x is -2/3 so the slope of the line of the given
equation is -2/3. You can also see by this comparison that the constant term is +20, and
so the y-intercept is +20.
.
So the answer to you problem is slope = -2/3 and y-intercept = +20 ... both of which you
can immediately identify by inspection.
.
Hope this helps you to understand the problem.
.

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Let x represent the unknown number.
.
Then one-half of the unknown number is x divided by 2 or
.
And one-sixth of the unknown number is x divided by 6 or
.
The problem tells you that one-half x is 3 more than one-sixth x. In equation form this can
be written as:
.

.
You can get rid of the denominators by multiplying both sides of this equation (all terms) by 6
to get:
.

.
Notice on the left side the 2 in the denominator divides into the 6 in the numerator
3 times. And on the right side the 6 in the denominator divides into the 6 in the numerator
1 time. These two reductions make the equation become:
.

.
Get rid of the x on the right side by subtracting x from both sides to get:
.

.
Finally, solve for x by dividing both sides by 2 to get:
.

.
Check the answer. One half of 9 is . One sixth of 9 is . Three more than
is
.
So does equal ???
.
Yes it does because if you divide 9 by 2 the answer is 4 1/2. And if you divide 27 by 6 the
answer is 4 3/6 which is also 4 1/2.
.
So, the answer checks. The unknown number you were to find is 9.
.
Hope this helps you to understand the problem.
.

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Since y varies directly as the square of x we can write the equation:
.

.
where k is just a constant that makes the equation true. Notice in this equation that if x
gets bigger, then y must also get bigger. This is what is meant by "y varies directly as"
.
We can find the value of k because we are told that when x = 2, then y = 32. Substitute these
two values into the above equation and you get:
.

.
Square the 2 on the right side and the equation is then:
.

.
Solve for k by dividing both sides by 4 to get:
.

.
Now we can return to the original equation and substitute 8 for k to get:
.

.
This the complete equation that relates x to y for this problem.
.
To find the value of y when x is 5 all we need to do is to substitute 5 for x in the equation
to get:
.

.
Squaring the 5 on the right side results in:
.

.
and the right side multiplies out to:
.

.
So the answer to your problem is that when x = 5, y equals 200.
.
Hope this helps you to understand the problem.
.

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Given to graph:
.

.
One way to work this problem is to recognize that you can manipulate linear inequalities
using the same rules as you do for solving equations, EXCEPT that if you multiply or divide
both sides of an inequality by a negative quantity, you must reverse the direction of the
inequality sign.
.
Knowing this, we can now get the given inequality into a slope-intercept form:
.

.
where m which is the multiplier of x is the slope of the graph, and b, the constant,
is the value on the y-axis where the graph crosses the y-axis.
.
Let's get the inequality into this form. Starting with the given:
.

.
Get rid of the 2x on the left side by subtracting 2x from both sides to get:
.

.
Solve for positive y by multiplying both sides of this inequality by -1. Don't forget the
rule that multiplying both sides by MINUS 1 makes it necessary to reverse the direction
of the inequality sign. So multiplying by -1 results in:
.

.
If this were an equation, what would the graph of this slope-intercept form look like? It
would have a slope of 2 and it would cross the y-axis at -4. In other words, it would look
like the following graph:
.

.
This graphed line shows all the (x, y) points in which the value of y is EQUAL TO 2x - 4.
.
But in your problem you need to identify all the (x, y) points that have a y-value that
is less than 2x - 4.
.
Those points will be all the (x, y) points that are below the line. [All the points that
have a y value greater than 2x - 4 are ABOVE the graphed line.]
.
So we identify the solution set by shading in the entire area of the graph that is below the
graphed line ... and do not include any points ON the graphed line. You can pick any
(x, y) point in that shaded area and you will find that its y value will be less than you
get if you take its x value and substitute it into 2x - 4.
.
For example, go into the shaded area (that is any place below the graphed line) and pick
any point in that area. Let's take the (x, y) point (+4, -3). If you substitute the value
of x (that is +4) into the function 2x - 4 you get 2(+4) - 4 = 8 - 4 = +4 as the value for
y. But the y value of the selected point is -3, so this selected point has a value of y that is
less than 2x - 4. You can try any number of points in the shaded area. You will get the same
result ... the y value for that point will be less than 2x - 4 where x is the x value of
that point. Therefore, any point in the shaded area satisfies:
.

.
which is just a form of the inequality you were given to graph.
.
The graph you are looking for is the shaded area below the graphed line shown in the above graph.
.
Hope this helps you to think your way through the problem so that you understand about graphing
linear inequalities.
.

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Let I represent the measure of the interior angle and E represent the measure of the exterior angle.
.
Note that an exterior angle is formed by just extending a side of the polygon. Once you see
that you can also see that an interior angle and its associated exterior angle are supplementary ...
meaning that their measures add to 180 degrees.
.
So, for this problem we can say that the supplementary equation is:
.
E + I = 180
.
The problem also tells you that the measure of each exterior angle (E) equals 25% (or 1/4) of
the measure of each interior angle (I). In equation form this relationship is:
.
E = (1/4)I
.
Substituting the right side of this equation for E in the supplementary equation results
in the supplementary equation becoming:
.
(1/4)I + I = 180
.
Get rid of the fraction by multiplying both sides of this equation (all terms) by 4 and
you get:
.
I + 4I = 720
.
The two terms on the left side sum to 5I so the equation is:
.
5I = 720
.
Solve for I by dividing both sides by 5 and you have:
.
I = 720/5 = 144
.
This means that the measure of each interior angle is 144 degrees. Since the measure of
the exterior angle is the supplement of the interior angle, this means that the measure of
each exterior angle is 180 - 144 = 36 degrees.
.
And as you correctly said, the sum of the measures of all the exterior angles must be 360 degrees.
So, the number of exterior angles must be 360 divided by 36 or 10. There are, therefore,
10 exterior angles, and since there is one exterior angle for each side, the polygon must
have 10 sides. The name for a 10-sided, 10-angled polygon is "decagon" and for this case
where the measures of all the angles of the polygon are equal, you can say that it is an
equiangular decagon, and since all sides must also be equal, the term used for an equiangular,
equal sided polygon is a "regular" polygon. This means we can also refer to the answer as a
"regular decagon."
.
Hope this clears up the problem for you.
.

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Given:
.

.
On the right side you can take the multiplier 3 to the inside of the log operator if you
make it an exponent. This makes the equation become:
.

.
Continuing on the right side, cube the 2 to get 8, making the equation become:
.

.
Notice that both sides have an operator of log to the base 10. For both sides of this equation
to be equal, the quantities that the log operator is applied to must be equal. This means
that
.
5 - x must equal 8. In equation form this is:
.
5 - x = 8
.
Get rid of the 5 on the left side by subtracting 5 from both sides to get:
.
-x = 3
.
Solve for x by multiplying both sides of this equation by -1 and the result is:
.
x = -3
.
That's the answer. You can check it by substituting -3 for x in the original equation you
were given. Start with the original equation of:
.

.
Substitute -3 for x and this equation becomes:
.

.
Simplify the left side by recognizing that 5 - (-3) = 5 + 3 = 8. This makes the equation become:
.

.
Using a calculator you can determine that the base 10 log of 8 is 0.903089987 and the
base 10 log of 2 is 0.301029995. Substituting these values results in:
.

.
If you multiply out the right side you will see that this equation is true, and therefore the
answer of x = -3 is correct.
.
Hope this helps you to work your way through this problem.
.

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The exterior angles of any "n-gon" always add up to 360 degrees. Exterior angles of a triangle?
360 degrees ... Of a rectangle? ... 360 degrees. Of a pentagon? 360 degrees. Of a hexagon? ...
360 degrees ... and so on ... The sum is always 360 degrees.
.
To get the measure of an exterior angle of a 15-gon, you have to assume that the 15-gon is
regular. Otherwise, you can't tell. All that you know is that the sum of the exterior angles
is 360 degrees and every exterior angle could have a different measure as long as the total is
still 360 degrees.
.
However, if you assume that the 15-gon is regular, then all the exterior angles (of which
there are 15) are equal. That being the case, you get the measure of each exterior angle
by dividing 360 degrees by 15 to find that each exterior angle is 24 degrees.
.
Hope this helps you to understand the concept of exterior angles of an n-gon.