SOLVING SYSTEMS OF LINEAR EQUATIONS BY ELIMINATION

Section 17.3

Using the Elimination Method

The elimination method utilizes the addition property of equality:

For example

If and , then .A B C D A C B D

If , then .A B A C B C or

Still adding the same thing to both sides since C = D.

&

Using the Elimination Method

Consider the system

7

5

x y

x y

A = B

C = D

A+C=B+D

Using the Elimination Method

Consider the system Consider the system

7

5

x y

x y

2 0 12x y 2 12x

6x

2 11

3 13

x y

x y

4 3 24x y

Uh oh. Cannot solve an

equation in two variables.

Adding worked because one variable had

opposite coefficients and thus added to zero and was eliminated.

Using the Elimination Method

For elimination to work, one of the variables must have opposite coefficients. If not, you can use the multiplication property to

change the coefficients.

This method is also called linear combination, or addition.

Using the Elimination Method

Consider the system

Solve using addition by eliminating either variable. Multiply the equations by any value that will produce

opposite coefficients on either variable. Must multiply one entire equation by the same value, but can use a different value for the other equation.

2 11

3 13

x y

x y

2 11

3 13

x y

x y

-3( ) ( )-3

3 6 33

3 13

x y

x y

5 20y 4y

Substitute to get x = 3

2 11

3 13

x y

x y

-2( )

( )-2

2 11

6 2 26

x y

x y

5 15x 3x

Substitute to get y = -4

OR