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Solving Systems Using
Elimination
Section 6-3
Goals
Goal
• To solve systems by adding
or subtracting to eliminate a
variable.
Rubric
Level 1 – Know the goals.
Level 2 – Fully understand the
goals.
Level 3 – Use the goals to
solve simple problems.
Level 4 – Use the goals to
solve more advanced problems.
Level 5 – Adapts and applies
the goals to different and more
complex problems.
Vocabulary
• Elimination Method
Elimination Method
Another method for solving systems of
equations is elimination. Like substitution, the
goal of elimination is to get one equation that
has only one variable. To do this by elimination,
you add the two equations in the system
together.
Remember that an equation stays balanced
if you add equal amounts to both sides. So,
if 5x + 2y = 1, you can add 5x + 2y to one
side of an equation and 1 to the other side
and the balance is maintained.
Elimination Method
Since –2y and 2y have opposite coefficients, the yterm is eliminated. The result is one equation that
has only one variable: 6x = –18.
When you use the elimination method to solve a
system of linear equations, align all like terms in the
equations. Then determine whether any like terms
can be eliminated because they have opposite
coefficients.
Solving Systems of
Equations
• So far, we have solved systems using
graphing and substitution. These notes show
how to solve the system algebraically using
ELIMINATION with addition and
subtraction.
• Elimination is easiest when the equations are
in standard form.
Elimination Procedure
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract the
equations.
Standard Form: Ax + By = C
Look for variables that have the
same coefficient.
Solve for the variable.
Step 4: Plug back in to find the
other variable.
Substitute the value of the variable
into the equation.
Step 5: Check your solution.
Substitute your ordered pair into
BOTH equations.
Example: Elimination by
Addition
x+y=5
3x – y = 7
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract the
equations.
They already are!
The y’s have the same
coefficient.
Add to eliminate y.
x+ y=5
(+) 3x – y = 7
4x = 12
x=3
Example: Continued
x+y=5
3x – y = 7
Step 4: Plug back in to find the
other variable.
Step 5: Check your solution.
x+y=5
(3) + y = 5
y=2
(3, 2)
(3) + (2) = 5
3(3) - (2) = 7
The solution is (3, 2). What do you think the answer would
be if you solved using substitution?
Your Turn:
Solve
y + 3x = –2
by elimination.
2y – 3x = 14
y + 3x = –2
2y – 3x = 14
3y + 0 = 12
3y = 12
Write the system so that like
terms are aligned.
Add the equations to
eliminate the x-terms.
Simplify and solve for y.
Divide both sides by 3.
y=4
Continued
y + 3x = –2
4 + 3x = –2
–4
–4
3x = –6
3x = –6
3
3
x = –2
(–2, 4)
Write one of the original
equations.
Substitute 4 for y.
Subtract 4 from both sides.
Divide both sides by 3.
Write the solution as an
ordered pair.
Your Turn:
Solve
3x – 4y = 10
by elimination.
x + 4y = –2
3x – 4y = 10
x + 4y = –2
4x + 0 = 8
4x = 8
4x = 8
4
4
x=2
Write the system so that like
terms are aligned.
Add the equations to
eliminate the y-terms.
Simplify and solve for x.
Divide both sides by 4.
Continued
x + 4y = –2
2 + 4y = –2
–2
–2
4y = –4
4y
–4
4
4
y = –1
(2, –1)
Write one of the original
equations.
Substitute 2 for x.
Subtract 2 from both sides.
Divide both sides by 4.
Write the solution as an
ordered pair.
More Elimination
• When two equations each contain the same
term, you can subtract one equation from
the other to solve the system.
• To subtract an equation add the opposite of
each term.
Example: Elimination by
Subtracting
4x + y = 7
4x – 2y = -2
Step 1: Put the equations in
Standard Form.
Step 2: Determine which
variable to eliminate.
Step 3: Add or subtract the
equations.
They already are!
The x’s have the same
coefficient.
Subtract to eliminate x.
4x + y = 7
(-) 4x – 2y = -2
3y = 9
Remember to
“keep-changey=3
change”
Example: Continued
4x + y = 7
4x – 2y = -2
Step 4: Plug back in to find the
other variable.
Step 5: Check your solution.
4x + y = 7
4x + (3) = 7
4x = 4
x=1
(1, 3)
4(1) + (3) = 7
4(1) - 2(3) = -2
Your Turn:
Solve
3x + 3y = 15
by elimination.
–2x + 3y = –5
3x + 3y = 15
–(–2x + 3y = –5)
3x + 3y = 15
+ 2x – 3y = +5
5x + 0 = 20
5x = 20
x=4
Add the opposite of each
term in the second
equation.
Eliminate the y term.
Simplify and solve for x.
Continued
3x + 3y = 15
3(4) + 3y = 15
12 + 3y = 15
–12
–12
3y = 3
y=1
(4, 1)
Write one of the original
equations.
Substitute 4 for x.
Subtract 12 from both sides.
Simplify and solve for y.
Write the solution as an
ordered pair.
Your Turn:
Solve
2x + y = –5
by elimination.
2x – 5y = 13
2x + y = –5
–(2x – 5y = 13)
2x + y = –5
–2x + 5y = –13
0 + 6y = –18
6y = –18
y = –3
Add the opposite of each
term in the second
equation.
Eliminate the x term.
Simplify and solve for y.
Continued
2x + y = –5
2x + (–3) = –5
2x – 3 = –5
+3
+3
2x
= –2
x = –1
(–1, –3)
Write one of the original
equations.
Substitute –3 for y.
Add 3 to both sides.
Simplify and solve for x.
Write the solution as an
ordered pair.
Joke Time
• What do you call a man with no arms and no legs
in a pool?
• Bob!
• What is T-Rex’s favorite number?
• Ate!
• How does the man in the moon cut his hair?
• Eclipse it!
Assignment
• 6-3 Part 1 Exercises Pg. 398: #6 – 20 even