Substitution Method(Windshield Wipers)
Linear Combinations (Elimination)
Useful technique for solving systems in which a variable has a coefficient of 1.
Useful when all variable have coefficients other than 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.
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 additive inverses.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
2x + 4(3x + 5) = 34
2x + 12x + 20 = 34
14x + 20 = 34
14x = 14
x = 1
y = 3(1) + 5
y = 3 + 5
y = 8
y = 3x + 5 2x + 4y = 34
x – 4y = -1
2x + 2y = 3
2(4y-1) + 2y = 3
8y – 2 + 2y = 3
10y – 2 = 3
10 y = 5
y =
x = 4 - 1
x = 2 - 1
x = 1
x - 4y = -1 2x + 2y = 3
x = 4y - 1
1
2
1
2
1
2
Decide which variable you want to eliminate.
I think I’ll choose to
eliminate the y variable.
3x – 5y = 14
2x + 4y = -20
3x – 5y = 14
2x + 4y = -20
12x - 20y = 56
10x + 20y = -100
22x = -44
x = -2
3(-2) – 5y = 14
-6 - 5y = 14
-5y = 20 y = -4
Decide which variable you want to eliminate.
I think I’ll choose to
eliminate the x variable.
2x + 7y = 48
3x + 5y = 28
2x + 7y = 48
3x + 5y = 28
6x + 21y = 144
-6x - 10y = -56
11y = 88
y = 8
3x + 5(8) = 28
3x + 40 = 28
3x = -12 x = -4
Decide which variable you want to eliminate.
I think I’ll choose to
eliminate the x variable.
4x + 3y = -19
6x + 5y = -32
4x + 3y = -19
6x + 5y = -32
12x + 9y = -57
-12x - 10y = 64
-y = 7
y = -7
6x + 5(-7) = -32
6x - 35 = -32
6x = 3
21x
12 , 7
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