Table of Contents. Inverse Operations. Click on a topic to go to that section. One Step Equations. Two Step Equations. MultiStep Equations. Variables on Both Sides. More Equations. Transforming Formulas. Inverse Operations. Return to Table of Contents. What is an equation?
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Inverse Operations
Click on a topic to go to that section.
One Step Equations
Two Step Equations
MultiStep Equations
Variables on Both Sides
More Equations
Transforming Formulas
An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent). Equations are written with an equal sign, as in
2 + 3 = 5
9 – 2 = 7
Equations can also be used to state the equality of two
expressions containing one or more variables.
In real numbers we can say, for example, that for any given value of x it is true that
4x + 1 = 14  1
If x = 3, then
4(3) + 1 = 14  1
12 + 1 = 13
13 = 13
When defining your variables, remember...
Letters from the beginning of the alphabet like a, b, c... often denote constants in the context of the discussion at hand.
While letters from end of the alphabet, like x, y, z..., are usually reserved for the variables, a convention initiated by Descartes.
Try It!
Write an equation with a variable and have a classmate identify the variable and its value.
An equation can be compared to a balanced scale.
Both sides need to contain the same quantity in order for it to be "balanced".
For example, 20 + 30 = 50 represents an equation because both sides simplify to 50.
20 + 30 = 50
50 = 50
Any of the numerical values in the equation can be represented by a variable.
Examples:
20 + c = 50
x + 30 = 50
20 + 30 = y
Why are we Solving Equations? both sides simplify to 50.
First we evaluated expressions where we were given the value of the variable and had to find what the expression simplified to.
Now, we are told what it simplifies to and we need to find the value of the variable.
When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true).
In order to solve an equation containing a variable, you need to use inverse (opposite/undoing) operations on both sides of the equation.
Let's review the inverses of each operation:
Addition Subtraction
Multiplication Division
There are four properties of equality that we will use to solve equations. They are as follows:
Addition Property
If a=b, then a + c=b + c for all real numbers a, b, and c. The same number can be added to each side of the equation without changing the solution of the equation.
Subtraction Property
If a=b, then ac=bc for all real numbers a, b, and c. The same number can be subtracted from each side of the equation without changing the solution of the equation.
Multiplication Property
If a=b, and c=0, then ac=bc for all real numbers ab, b, and c. Each side of an equation can be multiplied by the same nonzero number without changing the solution of the equation.
Division Property
If a=b, and c=0, then a/c=b/c for all real numbers ab, b, and c. Each side of an equation can be divided by the same nonzero number without changing the solution of the equation.
To solve for "x" in the following equation... solve equations. They are as follows:
x + 7 = 32
Determine what operation is being shown (in this case, it is addition). Do the inverse to both sides.
x + 7 = 32
 77
x = 25
In the original equation, replace x with 25 and see if it makes the equation true.
x + 7 = 32
25 + 7 = 32
32 = 32
For each equation, write the inverse operation needed to solve for the variable.
a.) y +7 = 14 subtract 7 b.) a  21 = 10 add 21
c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12
12
move
move
move
move
Think about this... solve for the variable.
To solve c  3 = 12
Which method is better? Why?
Kendra
Added 3 to each side of the equation
c  3 = 12
+3 +3
c = 15
Ted
Subtracted 12 from each side, then added 15.
c  3 = 12
12 12
c  15 = 0
+15 +15
c = 15
Think about this... solve for the variable.
In the expression
To which does the "" belong?
Does it belong to the x? The 5? Both?
The answer is that there is one negative so it is used once with either the variable or the 5. Generally, we assign it to the 5 to avoid creating a negative variable.
So:
×
What is the inverse operation needed to solve this equation?
7x = 49
1
Addition
A
Subtraction
B
Multiplication
C
D
Division
What is the inverse operation needed to solve this equation?
x  3 = 12
Addition
A
Subtraction
B
Multiplication
C
D
Division
To solve equations, you must work backwards through the order of operations to find the value of the variable.
Remember to use inverse operations in order to isolate the variable on one side of the equation.
Whatever you do to one side of an equation, you MUST do to the other side!
Examples: order of operations to find the value of the variable.
y + 9 = 16
 9 9 The inverse of adding 9 is subtracting 9
y = 7
6m = 72
6 6 The inverse of multiplying by 6 is dividing by 6
m = 12
Remember  whatever you do to one side of an equation, you MUST do to the other!!!
×
One Step Equations order of operations to find the value of the variable.
Solve each equation then click the box to see work & solution.
click to show
inverse operation
click to show
inverse operation
x  8 = 2
+8 +8
x = 6
2 = x  6
+6 +6
8 = x
click to show
inverse operation
click to show
inverse operation
x + 2 = 14
2 2
x = 16
7 = x + 3
3 3
4 = x
click to show
inverse operation
click to show
inverse operation
15 = x + 17
17 17
2 = x
x + 5 = 3
5 5
x = 2
One Step Equations order of operations to find the value of the variable.
3x = 15
3 3
x = 5
4x = 12
4 4
x = 3
25 = 5x
5 5
5 = x
click to show
inverse operation
click to show
inverse operation
x
(2)
= 10
(2)
2
x = 20
click to show
inverse operation
x
(6)
click to show
inverse operation
= 36
(6)
6
x = 216
click to show
inverse operation
TwoStep Equations order of operations to find the value of the variable.
Return to Table
of Contents
Sometimes it takes more than one step to solve an equation. Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.
This means that you undo in the opposite order (PEMDAS):
1st: Addition & Subtraction
2nd: Multiplication & Division
3rd: Exponents
4th: Parentheses
Whatever you do to one side of an equation, you MUST do to the other side!
Examples: Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.
3x + 4 = 10
 4  4 Undo addition first
3x = 6
3 3 Undo multiplication second
x = 2
4y  11 = 23
+ 11 +11 Undo subtraction first
4y = 12
4___4 Undo multiplication second
y = 3
Remember  whatever you do to one side
of an equation, you MUST do to the other!!!
×
Two Step Equations Remember that to solve equations, you must work backwards through the order of operations to find the value of the variable.
Solve each equation then click the box to see work & solution.
3x + 10 = 46
 10 10
3x = 36
3 3
x = 12
4x  3 = 25
+3 +3
4x = 28
4 4
x = 7
67x = 83
6 6
7x = 77
7 7
x = 11
2x + 3 = 1
 3 3
2x = 4
2 2
x = 2
9 + 2x = 23
9 9
2x = 14
2 2
x = 7
8  2x = 8
8 8
2x = 16
2 2
x = 8
Walter is a waiter at the Towne Diner. He earns a daily wage of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
12 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Solve the equation.
5x  6 = 56
13 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Solve the equation.
16 = 3m  8
Solve the equation. of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
x
2
14
 6 = 30
15 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Solve the equation.
5r  2 = 12
16 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Solve the equation.
12 = 2n  4
17 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Solve the equation.
 7 = 13
x
4
18 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Solve the equation.
+ 3 = 12
x
5

19 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
What is the value of n in the equation
0.6(n + 10) = 3.6?
0.4
A
5
B
4
C
D
4
20 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
In the equation
, n is equal to
8
A
2
B
1/2
C
D
1/8
Which value of x is the solution of the equation of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
21
1/2
A
2
B
2/3
C
D
3/2
22 of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Two angles are complementary. One angle has a measure that is five times the measure of the other angle. What is the measure, in degrees, of the larger angle?
MultiStep Equations of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
Return to Table
of Contents
Steps for Solving Multiple Step Equations of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
As equations become more complex, you should:
1. Simplify each side of the equation.
(Combining like terms and the distributive property)
2. Use inverse operations to solve the equation.
Remember, whatever you do to one side of an equation,
you MUST do to the other side!
Examples: of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
15 = 2x  9 + 4x
15 = 2x  9 Combine Like Terms
+9 +9 Undo Subtraction first
6 = 2x
2 2 Undo Multiplication second
3 = x
7x  3x  8 = 24
4x  8 = 24 Combine Like Terms
+ 8 +8 Undo Subtraction first
4x = 32
4___4 Undo Multiplication second
x = 8
×
Now try an example. Each term is infinitely cloned of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
so you can pull them down as you solve.
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 3
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
+ 6x
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
x = 9
Now try another example. Each term is infinitely cloned of $50, plus tips that are equal to 15% of the total cost of the dinners he serves. What was the total cost of the dinners he served if he earned $170 on Tuesday?
so you can pull them down as you solve.
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
6x
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
 5
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
+ x
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
44
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
x = 9
Always check to see that both sides of the equation are simplified before you begin solving the equation.
Sometimes, you need to use the distributive property in order to simplify part of the equation.
Distributive Property simplified before you begin solving the equation.
For all real numbers a, b, c
a(b + c) = ab + ac
a(b  c) = ab  ac
Examples simplified before you begin solving the equation.
5(20 + 6) = 5(20) + 5(6)
9(30  2) = 9(30)  9(2)
3(5 + 2x) = 3(5) + 3(2x)
2(4x  7) = 2(4x)  (2)(7)
Examples: simplified before you begin solving the equation.
5(1 + 6x) = 185
5 + 30x = 185 Distribute the 5 on the left side
5 5 Undo addition first
30x = 180
30 30 Undo multiplication second
x = 6
2x + 6(x  3) = 14
2x + 6x  18 = 14 Distribute the 6 through (x  3)
8x  18 = 14 Combine Like Terms
+18 +18 Undo subtraction
8x = 32
8 8 Undo multiplication
x = 4
×
Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down)
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
7x
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
)
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95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
95
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
x = 3
answer
Now show the distributing and solve...(each number/ symbol is infinitely cloned, so click on it and drag another one down)
)
)
)
)
)
)
)
)
)
)
)
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(
(
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(
(
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
2x
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
102
102
102
102
102
102
102
102
102
102
102
102
102
102
102
102
102
102
102
6
9
102
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
x = 4
answer
26 is infinitely cloned, so click on it and drag another one down)
Solve.
4 = 27y + 7  (15y) + 13
33 is infinitely cloned, so click on it and drag another one down)
What is the value of p in the equation
2(3p  4) = 10?
1
A
2 1/3
B
3
C
D
1/3
Variables on Both Sides is infinitely cloned, so click on it and drag another one down)
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of Contents
Remember... is infinitely cloned, so click on it and drag another one down)
1. Simplify both sides of the equation.
2. Collect the variable terms on one side of the equation. (Add or subtract one of the terms from both sides of the equation)
3. Solve the equation.
Remember, whatever you do to one side of an equation, you MUST do to the other side!
Example: is infinitely cloned, so click on it and drag another one down)
4x + 8 = 2x + 26
2x 2x Subtract 2x from both sides
2x + 8 = 26
 8 8 Undo Addition
2x = 18
2 2 Undo Multiplication
x = 9
What if you did it a little differently?
4x + 8 = 2x + 26
4x 4x Subtract 4x from both sides
8 = 2x + 26
26  26 Undo Addition
18 = 2x
2 2 Undo Multiplication
9 = x
Recommendation: Cancel the smaller amount of the variable!
×
Example: is infinitely cloned, so click on it and drag another one down)
6r  5 = 7r + 7  2r
6r  5 = 5r + 7 Simplify Each Side of Equation
5r 5r Subtract 5r from both sides (smaller than 6r)
r  5 = 7
+ 5 +5 Undo Subtraction
r = 12
×
Try these: is infinitely cloned, so click on it and drag another one down)
6x  2 = x + 13 4(x + 1) = 2x 2 5t  8 = 9t  10
x x 4x + 4 = 2x 2 5t 5t
5x  2 = 13 2x 2x 8 = 4t  10
+ 2 +2 2x + 4 = 2 +10 +10
5x = 15 4 4 2 = 4t
5 5 2x = 6 4 4
x = 3 2 2 = t
x = 3
1
2
Sometimes, you get an interesting answer. is infinitely cloned, so click on it and drag another one down)
What do you think about this?
What is the value of x?
3x  1 = 3x + 1
Since the equation is false, there is "no solution"!
No value will make this equation true.
move this
Sometimes, you get an interesting answer. is infinitely cloned, so click on it and drag another one down)
What do you think about this?
What is the value of x?
3(x  1) = 3x  3
Since the equation is true, there are infinitely many solutions! The equation is called an identity.
Any value will make this equation true.
move this
Try these: is infinitely cloned, so click on it and drag another one down)
4y = 2(y + 1) + 3(y  1) 14  (2x + 5) = 2x + 9 9m  8 = 9m + 4
4y = 2y + 2 + 3y  3 14  2x  5 = 2x + 9 9m  9m
4y = 5y  1 9  2x = 2x + 9 8 = 4
5y 5y +2x +2x No Solution
y = 1 9 = 9
y = 1 Identity
Mary and Jocelyn left school at 3:00 p.m. and bicycled home along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Define
t = Mary's time in hours
t + 0.25 = Jocelyn's time in hours
Jocelyn's distance
(rate time)
Mary's distance
(rate time)
equals
Relate
12t = 9(t+0.25)
Write
12t = 9(t + 0.25) along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
12t = 9t + 2.25
9t 9t
3t = 2.25
3 3
t = 0.75
It took Mary 0.75h, or 45 min, to get home.
×
34 along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Solve.
7f + 7 = 3f + 39
35 along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Solve.
h  4 = 5h + 26
36 along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Solve.
w  2 + 3w = 6 + 5w
37 along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Solve.
5(x  5) = 5x + 19
38 along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Solve.
4m + 8  2(m + 3) = 4m  8
39 along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
Solve.
28  7r = 7(4  r)
40 along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
In the accompanying diagram, the perimeter of ∆MNO is equal to the perimeter of square ABCD. If the sides of the triangle are represented by 4x + 4, 5x  3, and 17, and one side of the square is represented by 3x, find the length of a side of the square.
A
B
M
4x + 4
3x
N
17
D
5x – 3
C
O
More Equations along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
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of Contents
Remember... along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
1. Simplify each side of the equation.
Collect the variable terms on one side of the equation.
(Add or subtract one of the terms from both sides of the equation)
3. Solve the equation.
(Undo addition and subtraction first, multiplication and division second)
Remember, whatever you do to one side of an equation, you MUST do to the other side!
Examples: along the same bike path. Mary went at a speed of 12 mph and Jocelyn bicycled at 9 mph. Mary got home 15 minutes before Jocelyn. How long did it take Mary to get home?
x = 6
x = 6 Multiply both sides by the reciprocal of
x =
x = 10
2x  3 = + x
x  x Subtract x from both sides
x  3 =
+3 +3 Undo Subtraction
x =
3
5
5
3
5
3
3
5
5
3
×
30
3
14
5
14
5
1
5
There is more than one way to solve an equation with distribution.
Multiply by the reciprocal Multiply by the LCM
72
5
3
5
(3 + 3x) =
72
5
3
5
3
5
72
5
(3 + 3x) =
(3 + 3x) =
5
3
3
5
72
5
5
3
3
5
72
5
5
(3 + 3x) =
5
(3 + 3x) =
3(3 + 3x) = 72
9 + 9x = 72
+9 +9
9x = 81
9 9
x = 9
3 + 3x = 24
+3 +3
3x = 27
3 3
x = 9
Formulas show relationships between two or more variables. distribution.
You can transform a formula to describe one quantity in terms of the others by following the same steps as solving an equation.
Example: distribution.
Transform the formula d = r t to find a formula
for time in terms of distance and rate.
What does "time in terms of distance and rate" mean?
×
d = r t
r r
= t
Divide both sides by r
d
r
Examples distribution.
V = l wh Solve for w
V = w
l h
P = 2l + 2w Solve for l
2w 2w
P  2w = 2l
2 2
P  2w = l
2
×
Example: distribution.
To convert Fahrenheit temperature to Celsius, you use the formula:
C = (F  32)
Transform this formula to find Fahrenheit temperature in terms of Celsius temperature. (see next page)
5
9
Solve the formula for distribution.F
5
9
C = (F  32)
C = F 
+ +
C + = F
C + 32 = F
5
9
160
9
160
9
160
9
9
5
(
)
5
9
9
5
160
9
9
5
Solve the equation for the given variable. when given Area.
m p
n q
m p
n q
mq p
n
= for p
= (q)
=
2(t + r) = 5 for t
2(t + r) = 5
2 2
t + r =
 r  r
t =  r
(q)
5
2
5
2
51 when given Area.
The formula I = prtgives the amount of simple interest, I, earned by the principal, p, at an annual interest rate, r, over t years.
Solve this formula for p.
Irt
p =
A
Ir
t
p =
B
I
rt
p =
C
It
r
D
p =
Gm when given Area.
r
v2 =
A satellite's speed as it orbits the Earth is found using the formula . In this formula, m stands for the mass of the Earth. Transform this formula to find the mass of the Earth.
52
v2
G
 r
v2 – r
G
A
m =
m =
B
rv2  G
m =
C
rv2
G
m =
D
53 when given Area.
Solve for t in terms of s
4(t  s) = 7
7
4
t = + s
A
t = 28 + s
B
7
4
t =  s
C
7 + s
4
D
t =
56 when given Area.
Which equation is equivalent to 3x + 4y = 15?
A
y = 15 − 3x
y = 3x − 15
B
y = 15 – 3x
4
C
y = 3x – 15
4
D