Solving Linear Systems by Linear Combinations

7.3

• Objective 1 – Use linear combinations to solve a system of linear equations.

Steps to solve systems with linear combination.

1. Arrange equations with like terms in columns.

2. Combine like terms to get a variable to cancel.

3. Solve for one of the variables

4. Use the solution in step 3 to find the other variable.

Example3x – 2y = 12x + 2y = 4

Combine like terms incolumns. The y’s cancel

5x = 5 Divide by 55 5

x = 1 Now substitute 1 for x inone of the original equationsto solve for y.

2x + 2y = 4

2(1) +2y = 4

2 + 2y = 4

2y = 2

y = 1

Solution is (1,1)

EXAMPLE x + 4y = 23

– x + y = 2Combine like terms

5y = 25

y = 5 Use y = 5 to solve for x

x + 4y = 23

x + 4(5) = 23

x + 20 = 23

x = 3

Solution (3, 5)

EXAMPLE Using Multiplication

x – y = –5 x + 2y = 4

Nothing Cancels! We must multiply one of the equations by a value to get opposites

Multiply one equation by by -1

–1(x – y = –5) x + 2y = 4

– x + y = 5 x + 2y = 4

3y = 9

y = 3x – y = –5 x – 3 = – 5

x = – 2 Solution (-2, 3)