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