1. Solving Systems of Equations
Section 4.2

2. Useful when all variable have coefficients other than 1.
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.

3. Substitution Method (1, 8) y = 3x + 5 2x + 4y = 34 y = 3x + 5

4. Substitution Method (1, ) x – 4y = -1 2x + 2y = 3 x - 4y = -1
2(4y-1) + 2y = 3
8y – 2 + 2y = 3
10y – 2 = 3
x =
10 y = 5
x = 2 - 1
y =
x = 1
(1, )

5. Decide which variable you want to eliminate.
Linear
Combination
3x – 5y = 14
2x + 4y = -20
I think I’ll choose to eliminate the y variable.
Decide which variable you want to eliminate.

6. Linear Combination 4 5 (-2, -4) 3x – 5y = 14 2x + 4y = -20

7. Decide which variable you want to eliminate.
Linear
Combination
2x + 7y = 48
3x + 5y = 28
I think I’ll choose to eliminate the x variable.
Decide which variable you want to eliminate.

8. Linear Combination 3 -2 (-4, 8) 2x + 7y = 48 3x + 5y = 28

9. Decide which variable you want to eliminate.
Linear
Combination
4x + 3y = -19
6x + 5y = -32
I think I’ll choose to eliminate the x variable.
Decide which variable you want to eliminate.

10. Linear Combination 3 -2 4x + 3y = -19 6x + 5y = -32 12x + 9y = -57

11. Substitution Method Parallel Lines y = -2x - 6 6x + 3y = 11
- 18 = 11
Parallel Lines

12. Infinitely many solutions
Substitution
Method
x = 5y + 1
2x y = 2
2(5y + 1) y = 2
10y y = 2
2 = 2
Infinitely many solutions
same lines