Section 17.3
SOLVING SYSTEMS OF LINEAR EQUATIONS
BY ELIMINATION
Using the Elimination Method
 The elimination method utilizes the addition
property of equality:
If A  B, then A  C  B  C.
or
If A  B and C  D, then A  C  B  D.
Still adding the same thing to both sides since C = D.
 For example

&

 

Using the Elimination Method
 Consider the system
A
= B
x  y  7

x  y  5
C = D
A+C=B+D
Using the Elimination Method
 Consider the system
x  y  7

x y 5

2 x  0 y  12
2x  12
x6
Adding worked
because one
variable had
opposite
coefficients and
thus added to zero
and was
eliminated.
 Consider the system
 x  2 y  11

3 x  y  13

4 x  3 y  24
Uh oh. Cannot
solve an equation
in two variables.
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

 x  2 y  11

3 x  y  13
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.
OR
-3(
 x  2 y)(11)-3 3x  6 y
 33  x  2 y  11  x  2 y  11




3
x

y

13
3
x

y

13
-2(
6 x  2 y  26



3 x  y)  (13)-2 
5 y  20
5x  15
y  4
x3
Substitute to get x = 3
Substitute to get y = -4