Substitution Method
(Windshield Wipers)

Useful technique for solving
systems in which a variable
has a coefficient of 1.
Step 1: Solve one of the equations for
either one of its variables.
Step 2: Substitute the expression you
have for Step 1 into the
other equation and solve for
the remaining variable.
Step 3: Substitute the value from Step
2 back into the equation
from Step 1 and solve for the
second variable.
Step 4 : Check your solution in both of
the original equations.
Linear Combinations
(Elimination)

Useful when all variables
have coefficients other than
1.
Step 1: Arrange both equations so the like
terms line up in same column.
Step 2: Multiply one or both of the
equations by the same number
so the coefficients of one of the
variables are opposites.
Step 3: Add the equations together. One
of the variables should eliminate
because the coefficients will add
to zero.
Step 4: Solve for the remaining variable.
Step 5: Substitute the solution from Step 4
Into either of the original
equations and solve for the other
variable.
Step 6 : Check your solution in both of the
original equations.
y = 3x + 5
2x + 4y = 34
y = 3x + 5
2x + 4y = 34
2x + 4(3x + 5) = 34
2x + 12x + 20 = 34
y = 3(1) + 5
y=3+5
y=8
14x + 20 = 34
14x = 14
x = 1
x – 4y = -1
2x + 2y = 3
x - 4y = -1
2x + 2y = 3
x = 4y - 1
2(4y-1) + 2y = 3
8y – 2 + 2y = 3
x=
1 
4  2
x=2-1
x=1
1
10y – 2 = 3
10 y = 5
1
y =
2
1
2
3x – 5y = 14
2x + 4y = -20
Decide which variable
you want to eliminate.
I think I’ll
choose to
eliminate the y
variable.
3x – 5y = 14
2x + 4y = -20
56
3(-2) – 5y = 14
10x + 20y = -100
-6 - 5y = 14
12x - 20y =
22x
= -44
x = -2
-5y = 20
y = -4
2x + 7y = 48
3x + 5y = 28
Decide which variable
you want to eliminate.
I think I’ll
choose to
eliminate the x
variable.
2x + 7y = 48
3x + 5y = 28
6x + 21y = 144
3x + 5(8) = 28
-6x - 10y = -56
3x + 40 = 28
11y = 88
y = 8
3x = -12
x = -4
4x + 3y = -19
6x + 5y = -32
Decide which variable
you want to eliminate.
I think I’ll
choose to
eliminate the x
variable.
4x + 3y = -19
6x + 5y = -32
12x + 9y = -57
6x + 5(-7) = -32
-12x - 10y = 64
6x - 35 = -32
6x = 3
-y = 7
y = -7
x 

1
2
, 7 
1
2
y = -2x - 6
6x + 3y = 11
6x + 3(-2x - 6) = 11
6x - 6x - 18 = 11
- 18 = 11

x = 5y + 1
2x - 10y = 2
2(5y + 1) - 10y = 2
10y + 2 - 10y = 2
2= 2