Goal 1: Using algebraic methods to solve systems.
Substitution method
1.
Solve one of the equations for one of its variables
2.
Substitute the expression from step 1 into the other equation and solve for the other variable.
3.
Substitute the value from step 2 into the revised equation from step 1 and solve.

Example 1
Solve the linear system using the substitution method.
3x – y = 13
2x + 2y = -10

The linear combination method
1.
Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables.
2.
Add the revised equations from step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable.
3.
Substitute the value obtained in step 2 into either of the original equations and solve for the other variable.

Example 2
Solve the linear system using the linear combination method.
2x – 6y = 19
-3x + 2y = 10
Example 3
Solve the linear system using the linear combination method.
9x – 5y = -7
-6x + 4y = 2

Example 4
Solve the linear system. a.
9x – 3y = 15 b. 6x – 4y = 14
-3x + y = -5 -3x + 2y = 7

Example 5
A citrus fruit company plans to make 13.25 lb gift boxes of oranges and grapefruits. Each box is to have a retail value of $21.00. Each orange weighs
0.50 lb and has a retail value of $0.75, while each grapefruit weighs 0.75lb and has a retail value of
$1.25. How many oranges and grapefruits should be included in the box?