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First, get the equation into slope-intercept form: y = mx + b
4y + 5x = -12
subtract 5x from both sides
4y = -5x -12
divide both sides by 4
y = -5/4x -12/3
y = -5/4x -4
.
Recall,
the x-intercept is the point where y=0 and
the y-intercept is the point where x=0.
Thinking visually, the point (0, ??) will be on y-axis somewhere; likewise the point (??, 0) will be on the x-axis somewhere. Since we're drawing a line, it will cross at these points.
.
To find the y-intercept, simply set x=0:
the y-intercept is -4, which is at the point (0,-4)
.
Thinking about the slope-intercept equation, we know the slope is the coefficient of x.
In this case the slope is -5/4, which means it slopes downward from the upper left to the lower right.
We can find the x-intercept by setting y=0 and working it out.
0 = -5/4x -4
adding 5/4x to both sides
5/4x = -4
multiply both sides by 4
5x = -16
divide both sides by 5
x = -16/5
so the x-intercept is (-16/5, 0)
.
Now you have two points: (0,-4) and (-16/5,0).
With two points you can draw line.
Plot the two points and graph the equation.
.

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A good check is to find another point and verify it is on the line.
Say x= -4, which neatly cancels the denominator...
y = -5/4(-4) -4
y = 20/4 -4
y = 5 -4
y = 1
.
Is the point (-4,1) on the line?
Yes, it is.
.
Done.

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Start with your variables...
A = area of a triangle = 1/2*base*height
h = height of the triangle
b = base of the triangle
b = h + 6 :: the base is 6 cm > height
A = 20 cm^2
.
Substitute what we know...
20 cm^2 = 1/2*b*h
20 = 1/2(h+6)*h
multiply both sides by 2
40 = (h+6)*h = h^2 + 6h
subtract 40 from both sides
0 = h^2 + 6h - 40
so
h^2 + 6h - 40 = 0
factor
(h+10)(h-4) = 0
which means our solutions are
h = -10
h = 4
of course negative height is nonsense
which means h = 4 is the solution to test
.
substituting what we know about the base
b = h+6 = 10
.
checking the area
1/2*4*10 = 20?
2* 10 = 20
Yes.
.
Answer:
The height of the triangle is 4 cm.
The base of the triangle is 10 cm.
.
Done

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10a + 2b = 15
5a - 4b = 0
.
Multiply 2nd equation by 2 and then subtract
.
10a + 2b = 15
10a - 8b = 0
10b = 15
b = 1.5
.
5a = 4b
5a = 4*1.5 = 6
a = 1.2
.
Checking...does 10a + 2b = 15?
.
10a = 10*1.2= 12
2b= 2*1.5 = 3
12+3 = 15
Check.
.
Answer:
a + b = 1.2+1.5 = 2.7
.
Done.

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L = volume of liquid held by a large bottle
S = volume of liquid held by a small bottle
2S + L = 10 cups
or
L + 2S = 10
.
L - S = 4 cups
.
L + 2S = 10
L - S = 4
.
Subtracting the second equation from the first.
3S = 6
Divide both sides by 3
S = 2
.
Looking back, we see
L = S + 4 = 2 + 4 = 6
.
Checking these values to see if they're correct...
.
L + 2S = 10??
6 + 2*2 = 10??
Yes.
.
Answer:
The volume of the large bottle is 6 cups.
The volume of the small bottle is 2 cups.

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For jobs like this, look at it terms of the whole job being a unitary thing.
Carmen can do the job in 24 minutes, so she does the job at a rate of 1/24 of the total job per minute.
Carmen and Kim working together can do the job in 8 minutes, so their combined rate is 1/8 per minute.
Algebraically, their combined speeds can be shown as:

Multiplying through by 24x we have

Subtracting x from both sides


Dividing both sides by 2

Kim can do the job in 12 minutes working alone.
.
Checking our work, we were told that combined they did the job in 8 minutes.
In 8 minutes, Carmen can do of the job.
In 8 minutes, Kim can do of the job.
1/3 + 2/3 = 1 job.
.
Done

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There are 50 ways you can pick 1 item from a set of 50 items.
We're told that 4 of the 50 are defective, so 4/50 are bad.
So, picking 1 item, you would have a 4/50 or 1/12.5 or an 8% chance of drawing a bad one.
OR
You have a 46/50 chance of picking a good one, or 92%. That means the chance of drawing a bad one is 100% - 92% = 8%.
.
Done

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First, get it in y = mx+b format so that you can graph it.
5x + 4y = 20
subtract 5x from both sides
4y = -5x + 20
divide both sides by 4
y = -5/4x + 5
.
x = 0, y = 5 results in the point (0,5)
x = 4, y = 0 results in the point (4,0)
.
with two points you can graph a straight line
.

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Volume = 10 liters
Desired solution is 20% soap, so we assume that 80% is pure water.
Assuming we are measuring volumes, then the amount of pure soap in 10 liters of 20% soap is .2*10 = 2 liters of pure soap.
That means we add 8 liters of pure water to arrive at 10 liters.

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.



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The square root of 36 is either +6 or -6, BUT a negative width does not make sense, so we can start with w=6.
.

.
Checking the A

OK.
.
Answer:
The rectangle's length is 18; the width is 6.
.
Done

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mean = sum / count
mean = 6
count = 6
x = sum
6 = x / 6
x = 36
.
In 10 years, the sum will be x+10 = 36 + 10 = 46
mean = 46 / 6 = 7 2/3 = 7.67
.
Done

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First, get the equation into slope-intercept form: y = mx+b
-16x + 2y = -6
2y = 16x - 6
y = 8x - 3
.
But this line does NOT go through the point (16,8), as we can see on a graph.

In fact, y = 125 when x=16.
.
However, the general equation y = mx+b can be shifted with any constant 'b'.
8 = 8(16) + b
8 = 128 + b
b = -120
We can graph this as follows.

So the equation y = 8x - 120 will have slope = 8 and it will go through the point (16,8).
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We can check this answer by remembering the slope = rise / run = change in y / change in x.
The slope m = 8.
So if we move one point on the x-axis, there will be a change of 8 on the y-axis.
Given (16,8), we can predict that (15,0) will be on the line. (-1 on the x-axis leads to -8 on the y-axis)
And we can predict (17,16) will be on the line. (+1 on the x-axis leads to +8 on the y-axis)
Looking back at the second graph, we see that is the case.

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With 'solution' problems, you have to keep track of how much pure stuff you need.
This question is tricky because the question presents the complementary percentages.
Solution 1 is 70% water, so it is 30% pure acetic acid. We normally call that a 30% solution.
Solution 2 is 30% water, so it is 70% pure acetic acid. We normally call that a 70% solution.
The goal is 20 liters of a 60% acetic acid solution.
.
x = volume of 30% solution
20 - x = volume of 70% solution
.
.3x = amount of pure acetic acid
.7(20-x) = the other amount of pure acetic acid.
.
20 liters of 60% acetic acid will have .6*20 = 12 liters of pure acetic acid.
.
.3x + .7(20-x) = 12
.
.3x + 14 - .7x = 12
-.4x = -2
x = -2/-.4 = 5 liters of 30% solution
which means there is
15 liters of 70% solution.
.
checking to see if we have 12 L of pure acetic acid
.
.3*5 + .7*15 = 1.5 + 10.5 = 12
.
So, 5 liters of 30% acetic acid + 15 liters of 70% acetic acid results in 20 liters of 60% acetic acid solution.
.
Reviewing the question, we need to give the complementary answer in terms of percentage of water.
.
Answer:
5 liters of 70% water + 15 liters of 30% water = 20 liters of 40% water

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5p = 2n, where p = cost of pens and n = cost of notebooks
so,
n = 2.5p
.
p + 2n = 4.20, which is given
.
substituting
p + 2(2.5p) = 4.20
p + 5p = 4.20
6p = 4.20
p = .70 = 70 cents = cost of one pen
.
We also can calculate the cost of one notebook.
5p = 2n
5(.70) = 2n
3.50 = 2n
n = 1.75
.
Done

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5x -10y = -5
-10x -10y = 10
.
subtract the second equation from the first
.
15x = -15
x = -1
.
substituting to find y
.
5(-1) - 10y = -5
-5 -10y = -5
adding 5 to both sides
-10y = 0
so...
y = 0
.
Answer:
x = -1
y = 0
.
Done

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p = plane's speed in calm air
w = wind's speed
p + w = speed across the ground of the plane flying with the wind
p - w = speed across the ground of the plane flying against the wind
.
d = rt is the standard distance equation, where d=distance, r=rate, t=time
d = 800 miles is given
.
800 = 4(p+w) :: given that it takes 4 hrs with the wind to travel 800 miles
800 = 5(p-w) :: given that it takes 5 hrs against the wind to travel 800 miles
.
so we have
4(p+w) = 800
5(p-w) = 800
.
4p + 4w = 800
5p - 5w = 800
.
multiply the first equation by 5 and the second by 4
.
20p + 20w = 4000
20p - 20w = 3200
.
add them
.
40p = 7200
p = 180
.
substituting for p = 180, we can find w
.
4p + 4w = 800
divide by 4
p + w = 200
180 + w = 200
So,
w = 20
.
checking our work.
.
4(180+20) = 800??
4(200) = 800
Yes
.
5(180-20) = 800??
5(160) = 800
Yes.
.
Answer:
Plane flies at 180 mph in calm air.
Wind is blowing at 20 mph.
.
Done

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With an equilateral triangle, all sides are equal, so a = b = c, where a = one side's length, b = another side's length, and c = final side's length.
The perimeter of a triangle = a+b+c = 10.5
OR
3a = 10.5
a = 3.5
which means
b = 3.5
and
c = 3.5
.
done

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Given the uniform width of the border is 2.5 inches, then you will need to add 5 inches to the inner rectangle's length and width to determine the outer rectangle's area:
l + 5 = 7 + 5 = 12
w + 5 = 13 + 5 = 18
So, the outer rectangle's area = l*w = 12*18 = 216
.
The inner rectangle's are = 7 * 13 = 91
.
That means the area of the 'border' = 216 - 91 = 125
.
Done

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y = -4x
.
We know it is a straight line because there are no exponents.
To draw a straight line, you only need two points.
.
So, pick a couple of values of 'x' and compute the values of 'y'
for example...
x = 2, y = -8
x = -2, y = 8
.
The two point are thus: (2,-8), (-2, 8)
Plot them and draw the line.
.

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When solving number problems like these, we have to keep in mind the place values.
For example, the number 23 actually means 2*10 + 3*1.
Since the numbers are unknown in this word problem, we can call them 'xy'.
BUT keep in mind that they are just standing next to each other, they're not being multiplied.
The value would be 10x + y.
.
We are told some characteristics of the number 'xy'...
x + y = 14
.
That's easy, but the second statement is harder to set up.
.
Remember, the VALUE of 'xy' is 10x + y.
So, reversing the digits to be 'yx' would change the VALUE to 10y + x.
.
We're told
10y + x is 36 more than 10x + y, which algebraically is
10y+x = 10x+y+36
.
Rearranging this equation we have:
x + 10y = 10x + y + 36
x -10x + 10y - y = 36
-9x + 9y = 36
dividing by 9
-x + y = 4
.
Now we have to the two equations
x + y = 14
-x + y = 4
.
Adding we obtain
2y = 18
y = 9
.
Since x+y = 14, then x = 14-9 = 5
.
Checking...
xy = 59
.
yx = 95
.
Is 95 = 59 + 36??
Yes.
.
Answer:
The original number was 59.
.
Done

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volume of a cylinder = area of base * height = pi*r^2 * height
.
we are told volume = 91 and r=4
.
91 = pi(4^2) * h
91 = pi(16) * h
91 = 16pi * h
.
dividing by 16*pi
.
91/(16*pi) = h
.
so the height is about 17.8678 in.
.
done

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The area of a rectangle = length * width = l * w = 32 sq inches
.
l = w + 4 :: given
.
substituting
(w+4) * w = 32
w^2 + 4w = 32
w^2 + 4w - 32 = 0
.
factoring
(w+8)(w-4) = 0
.
so the solutions are w=-8 and w=4.
of course, a negative length is nonsense, so the only answer appears to be w=4
.
given l = w+4, then l= 4+4 = 8
.
checking the area
4*8 = 32??
yes
.
the perimeter of a rectangle = 2l + 2w = 2(8) + 2(4) = 16+8 = 24
.
Answer:
Length = 8
Width = 4
Perimeter = 24
.
done

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y = mx + b is the standard slope-intercept formula for a line
we can put the information given into two equations in slope-intercept format
R1 = .18x + 63, where x= miles driven
R2 = .09x + 126
.
we can graph the two lines to see where they cross
R1 = red line, which has a steeper slope
R2 = green line, which has a lower slope
.

.
it appears the lines cross at about x=700 (or 700 miles driven)
.
of course, we know R1 and R2 really are just other names for 'y'
.
y = .18x + 63
y = .09x + 126
.
multiplying both equations by 100 to remove decimals
100y = 18x + 6300
100y = 9x + 12600
.
rearranging to put into simultaneous equation format
.
-18x + 100y = 6300
-9x + 100y = 12600
,
subtracting the second equation from the first
-9x = -6300
.
dividing both sides by -9
x = 700
.
so we confirm that the two lines cross at x=700.
.
that means that at a distance driven 'x' of 700, the cost of the two rental plans is the same.
.
but after that point, the line with lower slope rises more slowly, which means the other plan (the one with the steeper slope) is more expensive and getting more expensive for every mile driven > 700.
.
Answer:
You need to drive more than 700 miles for Rate 2 to be less expensive than Rate 1.
.

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p = paddling rate in still water
c = current flow
p-c = speed going upstream
p+c = speed going downstream
d = rt, where d=distance, r=rate (or speed), and t= time
.
1.5*(p-c) = 6 miles = distance upstream
1*(p+c) = 6 miles = distance downstream
.
multiplying through
.
1.5p - 1.5c = 6
p + c = 6
.
dividing the first equation by 1.5
p - c = 4
.
solving as simultaneous equation, we can add them
p + c = 6
p - c = 4
2p = 10
p = 5 = paddling rate
.
substituting this value
p + c = 6
5 + c = 6
c = 1
.
checking by substituting in the other equation
.
1.5(p-c) = 6 ??
1.5(5-1) = 1.5*4 = 6
yes
.
so the answer is:
The rate of the current is 1 mph.
Cathy's paddling rate in still water is 5 mph.
.
done

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C = Cody's age
E = Evan's age
C = 2E :: Cody is twice as old as Evan.
Three years ago can be shown as:
C -3 = Cody's age 3 years ago
E -3 = Evan's age 3 years ago
C-3 + E-3 = 27 :: Given
C + E = 33
Substituting C = 2E
2E + E = 33
3E = 33
E = 11
Therefore
C = 2E = 22
How old were they 3 years ago?
C-3 = 19
E-3 = 8
Was the sum of their ages 27 back then?
19 + 8 = 27 ??
Yes.
Answer:
Cody's age is 22.
Evan's age is 11.
.
J = Jack's age
L = Lacy's age
J = 2L
In three years, their ages will be:
J+3 and L+3
Then the sum will be 54.
J+3 + L+3 = 54
J + L = 48
Substitue J = 2L
2L + L = 48
3L = 48
L = 16
J = 2L = 2(16) = 32
How old will they be in 3 years?
J = 35
L = 19
Will the sum of their ages be 54 then?
35 + 19 = 54 ??
Yes.
Answer:
Jack's age is 32 and Lacy's age is 16.
.
Done

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C = number of CDs
D = number of DVDs
.
Kate bought 3 CDs and 1 DVDs for $20:
3C + D = 20
.
Joel bought 2 CDs and 2 DVDs for $22:
2C + 2D = 22
.
Two equations and two unknowns can be solved using simultaneous equations.
.
3C + D = 20
2C + 2D = 22
.
Multiply the first equation by 2 and then subtract the second equation:
6C + 2D = 40
2C + 2D = 22
4C = 18
C = 4.50
.
Substituting to find D:
2C + 2D = 22
2(4.5) + 2D = 22
2D = 13
D = 6.50
.
Checking
3C + D = 20 ??
3(4.50) + 6.50 = 13.50 + 6.50 = 20
Yes.
.
Double checking...
2C + 2D = 22 ??
2(4.50) + 2(6.50) = 9 + 13 = 22
Yes.
.
Answer:
Used CDs cost $4.50 and used DVDs cost $6.50.
.
Done.

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The key to solving work problems like this is to determine how much of the job each per can do per unit of time. Then you solve how long it will take to total 1 job.
.
One person can do the job in 8 hr, so he does 1/8 of the job per hour.
Two people working together can do the job in 6 hr, so their combined speed is 1/6 of the job per hour.
.
1/8 + 1/x = 1/6
1/x = 1/6 - 1/8
1/x = 2/48 = 1/24
So, the second person takes 24 hr to do the job.
.
Checking how much they get done in 6 hours working together.
6 hr * 1/8 job/hr = 3/4 of the job
6 hr * 1/24 job/hr = 1/4 of the job
3/4 job + 1/4 job = 1 job
.
Answer:
The second person can do the job in 24 hr working alone.
.
Done

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S = Shelly's age
M = Michele's age
S = M + 3 :: Shelly is 3 years older than Michele.
.
"Four years ago" their ages will be S-4 and M-4. The sum of their ages was 67.
So, today we can say
S-4 + M-4 = 67
S + M -8 = 67
S + M = 75
.
Substitute S = M+3
.
(M+3) + M = 75
2M + 3 = 75
2M = 72
M =36
S = M+3 = 39
.
Four years ago their ages were:
S = 39 -4 = 35
M = 36 - 4 = 32
.
Checking if the sum 4 years ago = 67.
35 + 32 = 67
Yes.
.
Answer:
Shelly is 39, and Michele is 36.
.
Done

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x = the number
3x = three times the number
x+7 = 7 more than the number
2x - (-11) = twice the number minus -11 (if that is what you mean?)
.
3x + x+7 = 2x - (-11) = 2x+11
4x + 7 = 2x + 11
2x = 4
x = 2
.
Checking...
3x + x+7 = 3(2) + 2+7 = 6+9 = 15
2x + 11 = 2(2) + 11 = 15
.
So, the number is 2.

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L = length
W = width
P = perimeter = 2L + 2W = 2(L + W)
.
L = W+5
P = 74
.
2L + 2W = 74
substituting
2(W+5) + 2W = 74
2W + 10 + 2W = 74
4W = 64
W = 16
.
L = W+5 = 16+5 = 21
.
Checking
P = 74 ??
2(21) + 2(16) = 42 + 32 = 74
Yes.
.
Answer:
L = 21
W = 16
.
Done

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x = donut revenue
y = muffin revenue
x + y = 44000
so
y = 44000 - x
.
x = 4.5*y
.
substituting
x = 4.5(44000-x)
multiply though
x = 198000 - 4.5x
add 4.5x to both sides
5.5x = 198000
divide both sides by 5.5
x = 36000
.
y = 44000 - 36000 = 8000
.
Answer:
donut revenue = $36,000
muffin revenue = $8,000