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The function H(x) defines the height of a woman based on the length of her humerus.
H(x) = 2.75x + 71.48
.
x=33 is given.
.
H(x) = 2.75(33) + 71.48
H(x) = 90.75 + 71.48
H(x) = 162.23
.
Done

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A = cost of an apple
O = cost of an orange
B = cost of a banana
.
The apple costs the same as 2 oranges...
A = 2*O
or
O = A/2
.
An orange and a banana costs 10 cents more than an apple...
A + 10 = O + B
so
A = O + B - 10
and
O = A + 10 - B
and
B = A + 10 - O
.
Two oranges cost 15 cents more than a banana
2*O = B + 15
so
O = (B+15)/2
and
B = 2*O - 15
.
Now we solve these equations.
The key strategy is to get all of the costs in terms of the same fruit...
Since we have 3 equations that can be defined in terms of O, we can start with that.
.
O = A/2
O = A + 10 - B
O = (B+15)/2
.
If a=b and b=c, then a=c. So we know:
A/2 = A+10 - B
A = 2A + 20 - 2B
2B = A + 20
.
A/2 = (B+15)/2
A = B+15
B = A - 15
2B = 2A - 30
.
A+20 = 2A - 30
.
A = 50
.
Using A = 50 cents, we are able to find the other values by substituting back into the original equations.
.
A = 2*O
so
O = 25
.
B = A + 10 - O
B = 50 + 10 - 25
B = 35
.
Substituting back into the initial equations, we are able to check these values.
.
The apple costs the same as 2 oranges...
This statement was used to calculate the value of the oranges, so of course it checks.
.
An orange and a banana costs 10 cents more than an apple...
O + B = 25 + 35 = 60
That is 10 cents more than an apple, so it checks.
.
Two oranges cost 15 cents more than a banana
2*25 = 50, which is 15 cents more than a banaa.
.
These all check.
So we can conclude that in this case:
Apples cost 50 cents.
Oranges cost 25 cents.
Bananas cost 35 cents.
,
Done

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Start by setting up the variables you will need.
s = speed in still water = 9 mph
c = current
d = rt is the basic distance formula, where d=distance, r=rate, and t=time
.
Now translate the English into math.
.
Traveling downstream:
r = s+c The current adds to the boat's speed.
d=7 in time t
.
Traveling upstream:
r = s - c The current subtracts from the boat's speed.
d =3 in time t
.
We are told t=t so we can rearrange both equation to define 't'.
.
7 = (s+c) * t
.
Divide both sides by (s+c)
7/(s+c) = t
.
The other equation is:
3 = (s-c) * t
.
Divide both sides by (s-c)
3/(s-c) = t
.
Since both equations = t, we can use this fact to set them equal to one another
.
7/(s+c) = 3/(s-c)
.
Cross multiply
7(s-c) = 3(s+c)
.
7s - 7c = 3s + 3c
.
4s = 10c
.
Recall s = 9
.
4(9) = 36 = 10c
.
c = 3.6
The current runs at 3.6 mph
.
Next substitute back into the original equations to see if this answer is correct.
.
How long does it take to go 7 miles downstream?
7 = (9+3.6)t = 12.6t
t = 7/12.6 = .555555556
.
3 = (9-3.6)t = 5.4t
t = 3/5.4 = .555555556
,
So we can conclude the current of the river is 3.6 mph.
.
Done

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With coin problems you have to keep track of the number of coins and the value of the coins. Usually these create two linear equations.
.
Jana found she had nickels, dimes, and quarters with a total value of $1.90 = 190 cents.
The total value obviously is a value problem, so:
5n + 10d + 25q = 190
, where
n = number of nickels
d = number of dimes
q = number of quarters
.
But we need to figure out the number of coins, n, d, and q.
.
We are told in the problem:
d = 2q twice as many dimes as quarters
n = 1/2q half as many nickels as quarters
.
Substituting into the value equation, we have
.
5(1/2q) + 10(2q) + 25q = 190
.
Multiply by 2 to eliminate the fraction
5q + 40q + 50q = 380
.
95q = 380
.
q = 4
.
She has 4 quarters = 100 cents = $1.00
.
d = 2q = 2(4) = 8
.
She has 8 dimes = 80 cents = $.80
.
n = 1/2q = 1/2(4) = 2
.
She has 2 nickels = 10 cents = $.10
.
Adding up the dollar amounts, she has $1.90.
.
That checks.
Done.

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Step 1. Use some letter to represent the unknown quantity.
Step 2. Translate the English into mathematics and create an equation.
Step 3. Solve this equation.
Step 4. Check your work by substituting your results back into the original problem statement.
Step 5. Write your conclusion being sure answer the question asked.
.
STEP 1.
F = Father's age
M = Mary's age
.
STEP 2.
Father is 4 times as old as Mary: F = 4*M, where F=Father's age, and M=Mary's age.
Five years ago F was 7 times older than M.
F-5 = 7(M-5)
F-5 = 7M - 35
.
STEP 3.
substituting F = 4M
4M - 5 = 7M - 35
.
adding 5 to both sides
4M = 7M - 30
.
subtracting 7M from both sides
-3M = -30
.
dividing both sides by -3
M = 10
.
STEP 4.
Mary is 10 now.
Father is 4*10 = 40 now.
.
Five years ago:
Mary was 5.
Father was 35.
Was Father 7 times as old as Mary then?
Yes.
.
Checks.
STEP 5.
Mary is 10 years old now.
Father is 40 years old now.

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Brad's current age is: x
5 years ago, Brad's age was: x-5
At that time, his age was 2 more than 5/6*x.
What is x?
.
x-5 = 5/6*x + 2
.
Multiply through by 6 to eliminate the fraction
6x - 30 = 5x + 12
.
Subtract 5x from both sides
x - 30 = 12
.
Add 30 to both sides
.
x = 42
.
Five years ago, Brade's age was: x-5 = 37
5/6 of his current age is: 5/6x = 35
which is 2 less than 37.
Done

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Yes. There is no exponent involved.
You can show by graphing it that it is a straight line (linear).
To draw a straight line you only need two points and a straight edge to draw the line.
.
When x=0, y=-6. (0, -6)
.
When x=1/2, y=0. (1/2, 0)
.

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Work the algebra to get the equation into slope-intercept form: y = mx + b.
-3x -5y = -25
.
Add 3x to both sides
-5y = 3x -25
.
Divide both sides by -5.
y = -3/5x + 5
.
Slope is -3/5 so any line with slope -3/5 will be parallel.
y = -3/5x for example.
.
Any perpendicular will have negative inverse slope: 5/3.
y = 5/3x for example
.
The following graph illustrates:
red line: y=-3/5x+5
green line: y=-3/5x
blue line: y = 5/3x
.

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Slope-intercept form is: y = mx+b
m = 5 (given)
Since the slope = rise/run = change in 'y' divided by change in 'x', m=5 tells us that for a change of 1 in x we have a change of 5 in y.
.
One point is given: (4,4)
.
Substituting back into the equation, we have...
4 = 5(4) + b
4 = 20 + b
-16 = b
.
So the intercept appears to be -16.
.
The resulting equation is:
y = 5x - 16
.
Is it true for (4,4)?
Yes.
.
Is it true for another point?
.
Well, we can define another point based on knowing the slope.
.
m=5 means when we add 1 to x, the result is 5 being added to y.
Let's do just that: (4+1, 4+5) would do so.
.
Now we have two points: (4,4) and (5,9).
.
Does y= 5x -16 work for the point (5,9)?
.
9 = 5(5) - 16
9 = 25 - 16
True.
.
We also are informed by graphing the line.

.
Looking at the graph reminds us to check the point ( ?, 0).
0 = 5x - 16
5x = 16
x = 3.2
.
Inspecting the graph shows that looks right.
Done.

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time going upstream = 1.5 * time going downstream
distance = 72 miles
rate in still water (no current) = 30 mph
What is the current?
.
Recall d=rt, where d=distance, r=rate, and t=time.
.
We can set the upstream leg of the trip as:
72 = (30-x)*1.5
.
The downstream leg of the trip is:
72 = (30+x)*1
.
Since the distances are equal, we can set the right-hand sides to equal each other.
.
(30-x)*1.5 = 30+x
45 - 1.5x = 30 + x
.
Subtracting adding 1.5x to both sides...
45 = 30 + 2.5x
.
Subtracting 30 from both sides...
15 = 2.5x
so
2.5x = 15
.
Divide both sides by 2.5...
x = 15/2.5 = 6
.
So our proposed speed of the current = 6 mph.
.
Checking our work
.
The upstream leg would have a speed against the current of 24 mph. Traveling 72 miles would take 72/24 = 3 hr.
.
The downstream leg would have a speed assisted by the current of 36 mph. Traveling the 72 miles would take 72/36 = 2 hr.
.
Thus the roundtrip would be 5 hrs.
The total distance traveled would be 144 miles.
So the average speed would be be 144/5 = 28.8 mph
.
Done.

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We always seek ways to use the fewest unknowns when solving word problems.
.
We could call the two integers 'x' and 'y'. But we are told they are consecutive, so we can say:
x = x
y = x+1
That gives us only one unknown, 'x'.
.
We also are given some relationships:
the sum of two consecutive integers: x + x+1
is 27 more than 5 times the larger integer: 5(x+1) + 27
.
So,
x + x+1 = 5(x+1) + 27
.
Collecting and simplifying terms
2x + 1 = 5x + 5 + 27
2x + 1 = 5x + 32
.
Subtracting 1 from both sides
2x = 5x + 31
.
Subtracting 5x from both sides
-3x = 31
.
Dividing both sides by -3 shows that 'x' is NOT an integer, so there is no solution...
.
If only the setup had been 26 more instead of 27. (Or is that a typo?)
Then we would have had:
.
2x + 1 = 5x + 5 + 26
2x + 1 = 5x + 31
2x = 5x + 30
-3x = 30
x = -10
.
Then the two consecutive integers would have been: -10 and -9.
-10 + (-9) = -19
.
5*(-9) = -45
-45 + 26 = -19
.
Done.

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The key to solving "job" problems is to realize the job is 1 whole job that is divided into components that people do at different rates.
.
One painter paints the house in 6 hours, so his rate is 1/6 of the house per hour.
The other painter takes 9 hrs, so his rate is 1/9 of the house per hour.
.
Since we are not told otherwise, we assume they are working simultaneously: we assume they start and stop at the same time. So each of them works the same amount of time, which we will call 'x' hours.
.
First we set up time/rate for both painters: painter 1 works 'x' hrs at 1/6 of the whole job per hr; painter two works 'x' hrs at 1/9 of the whole job per hr:

.
The least common denominator is 54:

.
Cross multiply

.
Divide both sides by 15

Working together it takes them 3.6 hrs to paint the house.
.
Check by substituting back into the painter's rates to see if they total 1.


So working together they complete 1 whole job in 3.6 hr = 3 hr 36 min = 216 minutes, depending on the units the answer should be in.
.
Done.

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The cubic root of
The cubic root of
.
You can check by doing repeated multiplication:




.

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Turn each statement into an equation using the fewest possible unknowns.
.
L = length of the rectangle
.
Length is twice the width plus 5.
L = 2W + 5
.
P = perimeter of a rectangle.
P = 2L + 2W = 2(L + W)
P = 34 (given)
.
Substituting what we know...
34 = 2(L + W)
.
Divide both sides by 2...
17 = L + W
.
Substituting L = 2W+5...
17 = 2W + 5 + W
.
Collecting terms and subtracting 5 from both sides
12 = 3W
so
3W = 12
.
Dividing both sides by 3
W = 4
.
L = 2W+5
L = 2(4)+5
L = 8+5
L = 13
.
Checking the perimeter
P = 2(L+W) = 2(13+4) = 2(17) = 34
.
Checks.

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The elapsed time is 4 hr of which 2.5 hr was driving time.
The distance formula is d = rt, where d=distance, r=rate, t=time
We assume the distance to her sister's is the same as it is home (or she drives the same path both ways without any detours).
.
We can define the equation for driving to her sister's house:
d = 45(t)
.
Then the equation for driving home from her sister's house is:
d = 55(2.5-t)
.
In this way, we have defined the two unknown times in terms of a single unknown variable 't' and '2.5-t'.
.
Since d = d, we can say:
45t = 55(2.5-t)
45t = 137.5 - 55t
.
Adding 55t to both sides
100t = 137.5
.
Dividing both sides by 100
t = 1.375
.
So the time spend driving to her sister's house at 45 mph = 1.375 hr
The time spend driving home at 55 mph = 2.5 - 1.375 = 1.125 hr
.
How far does she travel at 45 mph?
d = 45 mph * 1.375 hr = 61.875 miles.
How far does she travel at 55 mph?
d = 55 mph * 1.125 hr = 61.875 miles.
.
So we've checked our work and are confident the distance is 61.875 = 61 7/8 miles.
.
We could be asked what her average speed was, too.
Was it simply (45+55)/2?
.
Well, we see that 2d = 2*61.875 = 123.75 miles
And it took 2.5 hr, so
t = d/r = 123.75/2.5 = 49.5.
So her average speed was 49.5.
.
Done.

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We know the basic formula for distance problems is: d = rt, where d=distance, r=rate, t=time.
We know the man drives to work at 50 mph.
We know he drives home at 30 mph.
We know the distance is the same both ways.
What we do not know are the times involved.
.
Let's assume he drives 30 miles to and from work.
At 50 mph it takes him 30/50 or .6 of an hour to drive to work. .6*60 = 36 minutes
At 30 mph it takes him 30/30 or 1 hr to drive home = 60 minutes
.
So the total elapsed time is 96 minutes to drive 60 miles.
96 minutes = 1.6 hr
.
Going back to d=rt
.
60 miles = r mph * 1.6 hr = 1.6r
.
Dividing both sides by 1.6
37.5 = r
r = 37.5
.
Checking our work...
How far will the car go in 1.6 hr at 37.5 mph?
60 miles
OK.

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The diagonal of the TV can be considered the hypotenuse of a right triangle. It can be solved using the Pythagorean formula:

.
Assuming L is the width of the TV set, which the problem calls "length", and H is the height of the screen.

.
Returning to the Pythagorean formula...

.

.
Substituting L = H+28




.
Squaring 52

.
Dividing both sides by 2

.
Subtracting 1352 from both sides

.
Simplifying

.
Can 960 be factored such that the two terms are 28 apart?
Yes. 48*20 = 960 and 48-20=28
.

.
So we have two candidate solutions: H= -48 and H = 20. Since a negative height is nonsense, then our suggested answer is:

.
Looking back to our defined equations,

.
Checking by using the Pythagorean formula:





.
So that checks just fine.
.
But is it an analog TV or an HDTV?
.
Analog TV has a ratio of width to height of 4:3. The picture is 4 units wide by 3 units high.
HDTV has a ration of 16:9. The picture is 16 units wide by 9 units high.
.
Our proposed TV set has a picture that is 48 wide by 20 high. That is a ratio of 48:20, or 24:10, or 12:5. This does not correspond to any real TV set. So perhaps a negative height would work when solving an "imaginary" TV problem. Hmmm...

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Setting up the equations is the key to solving word problems. Drawing a diagram is helpful, too.
.
L = W+4 (the length is 4 inches more than length (given))
.
L-2 (if 2 inches are taken from L) and added to the width (W+2) the figure becomes a square.
A square has 4 equal sides.
L-2 = W+2
.
We are told the area of the square is 196.
(L-2)(W+2) = 196
.
Substituting from L = W+4
(W+4 -2)(W + 2) = 196
(W+2)(W+2) = 196
.
Taking the square root
(W+2) = sqrt(196) = 14
.
Subtracting 2 from both sides
W = 12
and
L = W+4 = 16
.
Is the length 4 more than width? Yes.
If you subtract 2 from length and add 2 to width, and then multiply them, is the result 196? Yes.
Done.

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Percentage change is always calculated by

.

.
multiply by 100 to get percent
26.7% rounded to tenths
.

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Assuming it is a right triangle, the Pythagorean formula applies:

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.
Substitute


.
Subtract 88 from both sides

so

.
Can you factor this? Yes.
.

.
So our candidate solutions are L = -5.5 and L = 8.
.
A negative number for the side of the rectangle is nonsense. So our answer is L = 8.
.
Recall

so

.
With L = 8 and W = 11, the area = 88.
So we're done.

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rectangular garden. 4 feet longer than it is wide.
surrounded by a paved walkway 3 feet wide
the total area of the walkway is 432 square feet.
.
Garden facts:




.
Outside Dimensions of Walkway


Add six because the walkway is 3 ft wide on both ends.
.
The area of the walkway is 432, but you have to subtract the area of the garden.
.

.
Substitute for Lw and Ww

.
Multiply these


.
Collect terms


.
Divide both sides by 12

.
Simplify

.
Recall Lg = Wg + 4

.
The outside dimensions of the garden are the inside dimensions of the walkway:


.
The outside dimensions of the walkway are


.
Check the answer to ensure the area of the walkway is 432.
.
Garden:

.
Rectangle defined by the outside dimensions of walkway

.
Subtract the area of the garden to determine the area of the walkway

.
Done as an area problem.
.
Now let's look at a perimeter problem solution.
.
Looking at the basic facts as a perimeter problem, the walkway is 432 sq ft and we know it is 3 ft wide,
However, we have to visual how the walkway is fitted around the garden rectangle. It is best to draw a figure and label all the parts. You'll notice the walkway appears to follow the perimeter, which in fact it does.
BUT
The walkway also fills in the corners where the perimeter is only a right-angle point. In a sense, there is no length to add to the perimeter at the corner. There are 4 of these corners, each of which has to be 3x3 feet. So this amounts to 9*4 = 36 sq ft. That means that of the total 432 sq ft of walkway, only 396 sq ft align with a side of the rectangle. The rest is in the corners.
.
Recall that the perimeter (P) of a rectangle is:
P = 2(L+W) = 2L + 2W
.
Since the walkway beside the perimeter is 396 sq ft, and it is 3 ft wide, we can determine P by dividing 396 by 3.

.
To work with smaller numbers, we can redefine the perimeter equation by dividing by 2:

.
Thus we know the L+W = 66
.

.
Substituting W+4 for L:
.

.
Collecting terms and subtracting 4 from both sides


.
Divide both sides by 2 to eliminate the coefficient on W

.
Substituting back into the equation for L

.
Of course. the answer is the same as above, so this is another check.

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.
y-intercept is defined by f(0).
It is where the line crosses the y-axis.


.
x-intercept is defined by f(x) = 0.
It is where the line crosses the x-axis.
Recall

If x=4, then the entire right-hand side = 0, so that is an x-intercept.
If x=-3, then the entire right-hand side = 0, so that is another x-intercept.
.
The graph is useful to review. It is a parabola.

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=196 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 4, -3. Here's your graph:

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There is something missing from your statement. Please try again.

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The statements become the following equations.
2x + y = 18
4x - y = 12
.
2x + y = 18
Subtract 2x from both sides
y = -2x + 18
.
Substitute into the 2nd equation
4x - (-2x + 18) = 12
4x +2x - 18 = 12
Combine terms
6x -18 = 12
Add 18 to both sides
6x = 30
Divide both sides by 6
x = 5
.
Looking back, we recall:
2x + y = 18
4x - y = 12
.
2(5) + y = 18
10 + y = 18
Subtract 10 from both sides
y = 8
.
4(5)-8 = 12
True
.
So the two numbers are:
x = 5
y = 8

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The critical insight is that you have to get the equations in terms of accomplishing 1 job.
.
Sam can do the job in 20 min. So he can do the whole job at the rate of 1 job in 20 minutes.
So he does 1/20 of the job per minute.
.
Rob can do the job in 30 min. His rate is 1/30 of the job per minute.
.
Working together how does it take?
.
1/20*x + 1/30*x = 1 job done
, where x = minutes.
.
Multiply everything by 60 go get rid of the fractions.
3x + 2x = 60
5x = 60
.
Divide both sides by 12
x = 12
.
So our candidate answer is working together it will take them 12 minutes.
.
Let's check.
In 12 minutes, Sam does 12/20 of the job and Rob does 12/30 of the job.
12/20 = .6
12/30 = .4
Total = 1 job done.

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Arrange both equations in slope-intercept form.
.
The first equation is:

Subtract x from both sides

Divide both sides by 4

.
The second equation is:

Subtract 2x from both sides

Divide both sides by -6

.
By inspection, we know the two lines are not parallel because they do not have the same slope. So they cannot be the same line. We also can tell they are not perpendicular.
.
The two equations will intersect the points will be the same.
.

Multiply both sides by 12 to remove the fractions

Subtract 4x from both sides

Subtract 9 from both sides

Divide both sides by -7

.
Graph

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1.

subtract 36 from both sides

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=144 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 10, -2. Here's your graph:

.
.
2.

Add 7z to both sides and rearrange

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Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . The discriminant -47 is less than zero. That means that there are no solutions among real numbers. If you are a student of advanced school algebra and are aware about imaginary numbers, read on. In the field of imaginary numbers, the square root of -47 is + or - . The solution is Here's your graph:

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It depends on how men and women there are in the group of 9.