5-4 Elimination Using Multiplication
aka Linear Combination
Algebra 1
Glencoe McGraw-Hill
Linda Stamper
Often the equations are not ready for one variable to
cancel. You will need to create the opposites.
3x  8y  13
4x  4 y  12
Multiply one or both equations by a number to obtain
coefficients that are opposites for one of the variables.
33x  8y  13
13
4x  4 y  12
24x  4 y    12 2
Choose one of the variables to
create an opposite.
3x  8y  13
8x  8y  24
 11
11 x
x  1
Multiply by a number needed to
make one variable opposites.
Add your equation because
you now have opposites.
Substitute the solved value into
either of the original equations to
find the value for the other variable.
Yeah a handout. I
Write solutiondoas
an ordered pair.
not have to copy
these notes in my
notebook!
3  1  8y  13
 3  8y  13
8y  16
y 2
(–1,2)
33x  8y  13
13
4x  4 y  12
43x  8y   13 4
 34x  4 y    12   3
12x  32y  52
 12x  12y  36
44 y  88
While you can choose either of
the variables to make opposites,
choosing wisely may save you
some work. Here is the work if
the x variable is used to create
the opposite.
y 2
3 x  82  13
3x  16  13
 16   16
3x  3
x  1
(-1,2)
Solve the linear system using elimination.
Example 1
Example 2
3x  y  8
7 x  3y  8
3x  4 y  6
2x  5y  19
Example 1 Solve the linear system.
3xx  yy 88
3
7 x  3y  8
33x  y  83
9x  3y  24
7x  3y 
16x
8
 32
x 2
3 2  y  8
6 y 8
y 2
(2,2)
Example 2 Solve the linear system.
53x  4 y   65
33x  4 y  6
2x  5y  19
42x  5 y    19 4
15x  20y  30
8x  20y   76
23x
  46
x  2
3  2   4 y  6
 6  4y  6
4 y  12
y 3
(–2,3)
Example 2 Solve the linear system.
23x  4 y   62
33x  4 y  6
2x  5y  19
 32x  5y   19  3
6x  8y  12
 6x  15y  57
23y  69
y 3
3 x  43  6
3x  12  6
3x  6
x  2
(–2,3)
Solving A Linear System By Elimination
1) Arrange the equations with like terms in columns.
2) Multiply, if necessary, one or both equations by the
number needed to make one of the variables an
opposite.
3) Add the equations when one of the variables have
opposites. Then solve.
4) Substitute the value solved into either of the original
equations and solve for the other variable.
5) Check the ordered pair solution in each of the original
equations.
When solving a system by elimination, rearrange the terms
so that the corresponding variables are vertically stacked.
3x  6y  12
 x  3yy 66
3x  6y  12
3x  6y  12
 x  3y  6
3 x  3y   6(3)
Substitute the solved value into
either of the original equations.
 x  3(2)  6
Write answer as an ordered pair.
x66
x  0
x0
 3x  9y  18
15y  30
y 2
(0,2)
Solve the linear system using elimination.
Example 3
Example 4
3x  2y  8
2y  12  5x
 13  4x  3y
5x  2y  1
Example 5
3y  5x  15
6x  2y  18
Example 6 Write a linear system and then solve.
Five times the first number minus three times the
second number is six. Two times the first number
minus five times the second number is ten. Find
the numbers.
Assign labels.
Translate each sentence.
Solve the system.
Write a sentence to give the answer.
Example 3 Solve the linear system.
33x  2y  8 3x  2y  8  13x  2y  8  1
2y  12  5x
5x  2y  12
Rewrite in standard form.
3 (2)  2y  8
6  2y  8
 2  2y
1y
(2,1)
 3x  2y  8
5x  2y  12
2x  4
x 2
Example 4 Solve the linear system.
 13  4
4xx  3y
4x  3y  13
5x  2y  1
5x  2y  1
Rewrite in standard form.
 13  4 1  3y
 13  4  3y
 9  3 y
3y
(–1,3)
8x  6y  26
15x  6y  3
23x  23
x  1
Example 5 Solve the linear system.
3y  5x  15
6x  2y  18
 5x  3y  15
6x  2y  18
3y  5 3  15
3y  15  15
3y  0
0
y
3
y0
 3,0 
 10x  6y  30
18x  6y  54
8x  24
x  3
Example 6 Five times the first number minus 3 times the
second number is six. Two times the first number minus
five times the second number is 10. Find the numbers.
5x  3y  6  10 x  6y  12
2x  5y  10 10 x  25y  50
 19y  38
Translate each sentence.
y  2
Solve the system.
Let x = first number
Let y = second number
Write a sentence
to give the answer.
The numbers are -2 and 0.
5x  3  2  6
5x  6  6
5x  0
x0
Practice Problems
1) 2x  3y  4
 4x  5y  8
1) (2,0)
2) 6x  2y  2
 3x  3y  9
2) (1,–2)
3) 5x  4 y  3
2x  8y  2
1 1
3)  , 
3 3
4) Six times the first number plus two times the second
number is two. Four times the first number plus three
times the second number is eight. Find the numbers.
The numbers are -1 and 4.
5-A6 Page 276-278 #7–17,30,34-36,44-48.