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Solving Linear Systems by Elimination

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Solve systems of linear equations in two variables by combination/ elimination. Compare and choose an appropriate method for solving systems of linear equations. – PowerPoint PPT presentation

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Title: Solving Linear Systems by Elimination

1
Warm Up Simplify each expression. 1. 3x 2y
5x 2y 2. 5(x y) 2x 5y 3. 4y 6x 3(y
2x) 4. 2y 4x 2(4y 2x)
2x
7x
y
6y
Write the least common multiple.
5.
3 and 6
6
6.
4 and 10
20
7.
8.
6 and 8
2 and 5
24
10
2
Solving Linear Systems by Combination/ Elimination
3
Objectives
• Solve systems of linear equations in two
variables by combination/ elimination.
• Compare and choose an appropriate method for
solving systems of linear equations.

4
Another method for solving systems of equations
is combination. Like substitution, the goal of
combination is to get one equation that has only
one variable. To do this by combination, you add
or subtract 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.
5
Since 2y and 2y have opposite coefficients, the
y-term is eliminated. The result is one equation
that has only one variable 6x 18.
When you use the combination 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.
6
Solving Systems of Equations by Combination
COPY
Step 4
Write the answers from Steps 2 and 3 as an
ordered pair, (x, y), and check.
Note Later in this lesson you will learn how to
multiply one or more equations by a number in
order to produce opposites that can be eliminated.
7
COPY
3x 4y 10
Solve by elimination.
x 4y 2
Write the system so that like terms are aligned.
x 4y 2
Add the equations to eliminate the y-terms.
Step 2
4x 0 8
4x 8
Simplify and solve for x.
Divide both sides by 4.
8
COPY
Continued
Write one of the original equations.
2 4y 2
Substitute 2 for x.
Subtract 2 from both sides.
Divide both sides by 4.
Step 4 (2, 1)
Write the solution as an ordered pair.
9
Try this!
y 3x 2
Solve by combination.
2y 3x 14
Write the system so that like terms are aligned.
Add the equations to eliminate the x-terms.
3y 12
Simplify and solve for y.
Divide both sides by 3.
10
Try This! Continued
Write one of the original equations.
Step 3 y 3x 2
4 3x 2
Substitute 4 for y.
Subtract 4 from both sides.
Divide both sides by 3.
Write the solution as an ordered pair.
11
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.
12
COPY
Combination Using Subtraction
2x y 5
Solve by elimination.
2x 5y 13
2x y 5
Step 1
Add the opposite of each term in the second
equation.
(2x 5y 13)
Eliminate the x term.
Simplify and solve for y.
13
COPY
Continued
Write one of the original equations.
2x (3) 5
Substitute 3 for y.
2x 3 5
2x 2
Simplify and solve for x.
x 1
Write the solution as an ordered pair.
14
Try This!
3x 3y 15
Solve by combination.
2x 3y 5
Add the opposite of each term in the second
equation.
Eliminate the y term.
5x 0 20
Step 2
Simplify and solve for x.
15
Try This! Continued
Write one of the original equations.
Substitute 4 for x.
3(4) 3y 15
Subtract 12 from both sides.
Simplify and solve for y.
y 1
16
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17
Warm-up
• Write the least common multiple.
• 5 and 2
• 8 and 12
• 3 and 4
• 2 and 7
• 4 and 8
• 9 and 6

10
24
12
14
8
18
18
In some cases, you will first need to multiply
one or both of the equations by a number so that
one variable has opposite coefficients. We call
this the Elimination Method. This will be the new
Step 2.
19
COPY
Solving Systems of Equations by Elimination Using
Multiplication
Write the system so that like terms are aligned.
Step 1
Multiply one or both of the equations by a number
to obtain coefficients that are opposites for one
of the variables.
Step 2
Eliminate one of the variables and solve for the
other variable using the new equations your
created in Step 2.
Step 3
Substitute the value of the variable into one of
the original equations and solve for the other
variable.
Step 4
Write the answers from Steps 2 and 3 as an
ordered pair, (x, y), and check.
Step 5
20
COPY
Elimination Using Multiplication First
Solve the system by elimination.
x 2y 11
3x y 5
Multiply each term in the second equation by 2
to get opposite y-coefficients.
Add the new equation to the first equation.
7x 0 21
Simplify and solve for x.
21
COPY
Continued
Write one of the original equations.
Substitute 3 for x.
3 2y 11
Subtract 3 from each side.
Simplify and solve for y.
y 4
Write the solution as an ordered pair.
22
COPY
Elimination Using Multiplication First
Solve the system by elimination.
5x 2y 32
2x 3y 10
Multiply the first equation by 2 and the second
equation by 5 to get opposite x-coefficients
19y 114
Step 2
Simplify and solve for y.
y 6
23
COPY
Continued
Write one of the original equations.
2x 3(6) 10
Substitute 6 for y.
2x 18 10
Subtract 18 from both sides.
x 4
Simplify and solve for x.
24
Try This!
Solve the system by elimination.
3x 2y 6
x y 2
Multiply each term in the second equation by 3 to
get opposite x-coefficients.
Add the new equation to the first equation.
25
Try This! Continued
Write one of the original equations.
x 3(0) 2
Substitute 0 for y.
x 0 2
Simplify and solve for x.
x 2
x 2
26
Try this!
Solve the system by elimination.
2x 5y 26
3x 4y 25
Multiply the first equation by 3 and the second
equation by 2 to get opposite x-coefficients
(2)(3x 4y 25)
6x 15y 78
(6x 8y 50)
Simplify and solve for y.
y 4
27
Try this! Continued
Write one of the original equations.
2x 5(4) 26
Substitute 4 for y.
Subtract 20 from both sides.
Simplify and solve for x.
x 3
28
Lesson Quiz
Solve each system by elimination. 1. 2. 3.

2x y 25
(11, 3)
3y 2x 13
3x 4y 18
(2, 3)
x 2y 4
2x 3y 15
(3, 7)
3x 2y 23