Solving Linear Systems Algebraically

with Substitution

Section 3-2

Pages 160-1-67

Objectives

• I can use the substitution method to solve equations

• I can solve word problems using Substitution

Substitution Method

• Goal

• 1. Isolate one variable in one equation

• 2. Substitute into the other equation(s)

• AWAYS pick the easiest equation to isolate.

Which Equation to Isolate

82

932

yx

yx

842

96

yx

xy

1042

125

yx

yx84

34

yx

yx

2 8x y 6 9y x

5 12y x 4 3

4 8

y x

x y

Example 1

2 5 7 4 2x y x y

4 2x y 2( 4 2) 5 7y y

8 4 5 7y y

3 3y 1y

4( 1) 2x

6x

(6, 1)

What does it mean?

• When we found the solution (6, -1)

• What does that really mean???

• Intersection of the 2 graphs!!

2 72 5 7

5 51 1

4 24 2

x y y x

x y y x

1 2 63 4 5 7 8 9 10

4

3

2

7

56

8

9

x-axis

y-axis

0

1-2-6 -3-4-5-7-8-910

-4

-3

-2

-1

-7

-5

-6

-8

-9

0

-1

y=-2/5x+7/5

y=-1/4x+1/2

(6, -1)

Example 2

3 2 3 3 3x y x y

3 3y x 3 2( 3 3) 3x x

3 6 6 3x x

9 3x 1

3x

13( ) 3

3y

2y 1

,23

Your Turn

• Solve the following system of equations using substitution:

124

623

yx

yx)3,0(:Solution

Other Methods

• Remember, the solution to a system of equations if an Ordered Pair

• You know 2 other methods to check your answers:– Graphing to find the intersection– Graphing Calculator and asking for the

intersection (2nd, Trace, Intersection, E, E, E)

Solution Types

Remember there are 3 types of solutions possible from a system of equations!

No Solution vs Infinite

• How will you know if you have No Solution or Infinite Solutions when solving by Substitution??

Remember Back to Solving Equations

No Solution• Variables are gone and

you get this:

• 2x + 3 = 2x – 4• 3 = -4• This is not possible, so

• No Solution

Infinite Solutions• Variables are gone and

you get this:

• 2x + 3 = 2x + 3• 3 = 3• This is always true, so

• Infinite Solutions

Word Problems

• When solving a word problem, consider these suggestions

• 1. Identify what the variables are in the problem

• 2. Write equations that would represent the word problem, looking for key words

• Sum, difference, twice, product, half, etc…

Example 1

• GEOMETRY: The length of a rectangle is 3 cm more than twice the width. If the perimeter is 84 cm, find the dimensions.

Variables:

Length (L)

Width (W)

Equations:

L = 2W + 3

2L + 2W = 84

Now, solve by substitution

Example 2

• Melissa has 57 coins in dimes and nickels. The total value of the coins is $4.60. How many coins of each kind does she have?

Nickels (N)

Dimes (D)

Equations:

N + D = 57

10D + 5N = 460

Now, solve by substitution

Example 3

• At a recent movie, adult tickets were $4.50 and student tickets were $2.50. During opening night a total of 300 tickets were sold earning $1130. How many of each ticket type were sold?

Adult Ticket (A)

Student Ticket (S)

Equations:

A + S = 300

4.50A + 2.50S = 1130

Now, solve by substitution

Homework

• Substitution Worksheet