Download Solving Systems Using Elimination

Solving Systems Using
Elimination
Elimination
When neither equation is in the slopeintercept form (y =), you can solve the
system using elimination.
 You can add equations to eliminate a
variable.
 Look for like terms that are opposites of
each other (will add to zero).

Example 1
5x – 6y = -32
 3x + 6y = 48
 Make sure like terms are lined up with each
other.
 Look for like terms that are opposites (in
this case the –6y and +6y are opposites).
 Add all the like terms (5x + 3x = 8x) (-6y +
6y = 0) (-32 + 48 = 16)

Example 1
Now you have “eliminated” the y term and
have a one-variable equation to solve.
 8x = 16
 So, x = 2
 You have the first half of your ordered pair.
Plug in 2 for x in one of the equations to
find y.
 5(2) – 6y = -32

Example 1
10 – 6y = -32 Subtract 10 from both sides.
 -6y = -42 Divide both sides by –6
y=7
 Check by replacing x with 2 and y with 7 in
the second equation.
 3(2) + 6(7) = 48
 6 + 42 = 48 That’s true, so the solution is
(2,7)

Example 2
x – y = 12
 x + y = 22
 The y’s are opposites, so add like terms
 (x + x = 2x, -y + y = 0, and 12 + 22 = 34)
 Now your equation is 2x = 34. Divide both
sides by 2.
 x = 17

Example 2
Replace x with 17 in the first equation to
find y. 17 – y = 12
 Subtract 17 from both sides. -y = -5
 Divide both sides by –1. y = 5
 Replace x with 17 and y with 5 in the
second equation to check. 17 + 5 = 22
 That is true, so the solution is (17, 5)

Not Always So Easy
Sometimes there are not like terms that will
add to zero (eliminate).
 You can multiply or divide all the terms by
any number (except zero) to make
opposites.

Example 3
3x + 4y = -10
 5x – 2y = 18
 There are no opposite like terms. However,
if I multiply the second equation by 2, the
y’s will be opposites. (The first equation
will stay the same.)
 3x + 4y = -10
 10x – 4y = 36

Example 3
Now add the like terms.
 13x = 26
 Solve. x = 2
 Replace x in the first equation with 2 to find
y.
 3(2) + 4y = -10
 6 + 4y = -10 Subtract 6 from both sides.

Example 3
4y = -16 Divide both sides by 4
 y = -4
 Check by replacing x with 2 and y with –4
in the second equation.
 5(2) – 2(-4) = 18
 10 + 8 = 18. This is true.
 The solution is (2, -4)

Example 4
7x – 12y = -22
5x – 8y = -14
 I choose to get rid of the x’s. So I will
multiply the top equation by 5 and the
bottom equation by -7
 35x – 60y = -110
 -35x + 56y = 98
 Now add the equations.

Example 4
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
-4y = -12
Y=3
That is the second member of the ordered pair.
Now find x by replacing y in one of the equations
with 3 and solve for x.
7x – 12(3) = -22
7x – 36 = -22
7x = 14
X=2
(2, 3)
Try these…
2x + 7y = 31
5x – 7y = -45
 x – 6y = 2
6x + 6y = 12
 2x + 5y = 34
x + 2y = 14
 x + 6y = 20
x + 2y = 12


(-2, 5)

(2, 0)

(2, 6)

(8, 2)