Using Algebraic Methods

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Using Algebraic Methods to Solve Linear Systems
3-2
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
2
Warm Up Determine if the given ordered pair is an
element of the solution set of
2x y 5
.
3y x 6
no
2. (1, 1)
1. (3, 1)
yes
Solve each equation for y.
3. x 3y 2x 4y 4
y x 4
4. 6x 5 y 3y 2x 1
y 2x 3
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Objectives
Solve systems of equations by substitution.
Solve systems of equations by elimination.
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Vocabulary
substitution elimination
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The graph shows a system of linear equations. As
you can see, without the use of technology,
determining the solution from the graph is not
easy. You can use the substitution method to find
an exact solution. In substitution, you solve one
equation for one variable and then substitute
this expression into the other equation.
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Example 1A Solving Linear Systems by Substitution
Use substitution to solve the system of equations.
Step 1 Solve one equation for one variable. The
first equation is already solved for y y x 1.
Step 2 Substitute the expression into the other
equation.
x y 7
x (x 1) 7
Substitute (x 1) for y in the other equation.
2x 1 7
Combine like terms.
2x 8
x 4
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Example 1A Continued

• Step 3 Substitute the x-value into one of the
  original equations to solve for y.

y x 1
y (4) 1
Substitute x 4.
y 3
The solution is the ordered pair (4, 3).
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Example 1A Continued
Check A graph or table supports your answer.
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Example 1B Solving Linear Systems by Substitution
Use substitution to solve the system of equations.
Method 2 Isolate x.
Method 1 Isolate y.
2y x 4
2y x 4
First equation.
x 4 2y
Isolate one variable.
3x 4y 7
3x 4y 7
Second equation.
3(4 2y) 4y 7
Substitute the expression into the second
equation.
12 6y 4y 7
3x 2x 8 7
12 10y 7
5x 8 7
Combine like terms.
10y 5
5x 15
x 3
First part of the solution
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Example 1B Continued

• Substitute the value into one of the original
  equations to solve for the other variable.

Method 1
Method 2
Substitute the value to solve for the other
variable.
2y (3) 4

1. x 4

1 x 4
2y 1
x 3
Second part of the solution
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Check It Out! Example 1a
Use substitution to solve the system of equations.
Step 1 Solve one equation for one variable. The
first equation is already solved for y y 2x
1.
Step 2 Substitute the expression into the other
equation.
3x 2y 26
Substitute (2x 1) for y in the other equation.
3x 2(2x1) 26
3x 4x 2 26
Combine like terms.
7x 28
x 4
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Check It Out! Example 1a Continued

• Step 3 Substitute the x-value into one of the
  original equations to solve for y.

y 2x 1
y 2(4) 1
Substitute x 4.
y 7
The solution is the ordered pair (4, 7).
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Check It Out! Example 1a Continued

• Check A graph or table supports your answer.

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Check It Out! Example 1b
Use substitution to solve the system of equations.
Method 1 Isolate y.
Method 2 Isolate x.
2x 2 y
2x 2 y
First equation.
y 2x 2
Isolate one variable.
5x 6y 9
5x 6y 9
Second equation.
Substitute the expression into the second
equation.
5x 6(2x 2) 9
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Check It Out! Example 1b Continued
Method 1 Isolate y.
Method 2 Isolate x.
Combine like terms.
5x 6(2x 2) 9
5x 12x 12 9
10 5y 12y 18
7x 21
10 7y 18
7y 28
x 3
y 4
First part of the solution.
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Example 1b Continued

• Substitute the value into one of the original
  equations to solve for the other variable.

5(3) 6y 9
5x 6(4) 9
Substitute the value to solve for the other
variable.
5x (24) 9
15 6y 9
6y 24
5x 15
x 3
y 4
Second part of the solution
By either method, the solution is (3, 4).
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Lesson Quiz
Use substitution or elimination to solve each
system of equations.
3x y 1
5x 4y 10
1.
2.
y x 9
3x 4y 2
(2, 7)
(6, 5)
3. The Miller and Benson families went to a
theme park. The Millers bought 6 adult and 15
children tickets for 423. The Bensons bought
5 adult and 9 children tickets for 293. Find
the cost of each ticket.
adult 28 childrens 17