# Process for Solving Linear Equations for the y Variable ```Process for Solving Linear Equations for the y Variable
When working with linear equations, we will often be given equations of the form
y  mx  b (m and b being constants), where the y-variable is already isolated. This
makes our process of graphing much easier.
However, equations are sometimes given to us in the form Ax  By  C (A, B, and C
being constants), and we would like to solve for y. This is not a difficult process as long
as we remember the basic principals of solving equations.
1. If any fractions or decimals occur in the equation, multiply through by the LCD to
clear denominators and decimals.
2. Using the addition principal, add or subtract the entire Ax-term to move it to the
other side of the equation.
3. Using the multiplication principal, divide both sides of the equation by B in order
to isolate y. Be certain that BOTH the C and the Ax terms get divided by B!
Notice in step two we will be using addition or subtraction. In step three we will be using
division. We are &quot;undoing&quot; the equation so the various parts of the equation &quot;let go&quot; of
stronger connections (multiply/divide). Think of it as moving backwards through the
order of operations.
Order of Operations
P
E
Simplify
Solve
MD
AS
EXAMPLE #1: Solve 3x  2 y  7 for y
3x  2 y  7
 3x
 3x
 2 y  3x  7

2
2
 3x
7
y

2 2
3
7
y  x
2
2
Subtracting 3x from both sides
Dividing both sides by -2
Be sure BOTH terms are divided!
EXAMPLE #2. Solve 
1 
 2
15  x  y   315
5 
 3
 2 
1 
15  x   15 y   45
 3 
5 
 10 x  3 y  45
 10 x
 10 x
3 y 10 x  45

3
3
10 x 45
y

3
3
10
y  x  15
3
2
1
x  y  3 for y
3
5
Multiply both sides by LCD (15)
Distribute 15 thru: every term must
get multiplied by the 15!
Divide both sides by 3
Be sure BOTH terms get divided
EXAMPLE #3. Solve 3x  y  12 for y
3 x  y  12
 3x
 3x
 y  3 x  12

1
1
 3 x 12
y

1 1
y  3 x  12
Subtract 3x from both
sides
Divide by -1 to get y