even though he knows you are slightly cracked.”

"A true friend is someone who thinks you are a good egg 

Warm UpHow many solutions to the system shown in each graph?

x

y

x

y

x

y

(1)

(2)

(3)

one solution

no solution

infinitely many

solutions

5-2A Solving Linear Systems by Substitution

Algebra 1 Glencoe McGraw-Hill Linda Stamper

Two or more linear equations in the same variable form a system of linear equations, or simply a linear system.

x + y = 5 Equation 1 2x – 3y = 3 Equation 2

A solution of a linear system in two variables is an ordered pair that makes each equation a true statement.

In the previous lesson you solved a linear system by using a graph.

The point where the graphs of each equation intersect is the ordered pair solution.

Point of intersection

(–1,–3)

x

y

•

There are several ways to solve a linear system without using a graph. In this lesson you will study an algebraic method known as the substitution method.

1) Choose one of the equations and isolate one of the variables.

2) Substitute the expression from Step 1 into the other equation and solve.

3) Substitute the solved variable from Step 2 into either of the original equations and solve. Write the answer as an ordered pair.

4) Check the ordered pair solution in each of the original equations.

Solving A Linear System By Substitution

2y3x2 1yx

First choose one equation and isolate one of the variables. You will get the same solution whether you solve for x first or y first.

You should begin by solving for the variable that is easier to isolate.

Which of the above equations would be easier to isolate one of the variables?

1xy Here is a linear system:

Which equation would you choose to isolate the variable?Name the variable you would solve for first.8y4x2 and 9 yx3

y for solve

11y3x and 33y5x2

x for solve

0y3x and 10y2x

equation either in x for solve

1) Choose one of the equations and isolate one of the variables.

2) Substitute the expression from Step 1 into the other equation and solve.

3) Substitute the solved variable from Step 2 into either of the original equations and solve. Write the answer as an ordered pair.

4) Check the ordered pair solution in each of the original equations.

Solving A Linear System By Substitution

2 ) ( x2

Solve the linear system using the substitution method.Choose one equation and isolate one of the variables.

1xy Substitute the expression into the other equation and solve. 1x

3x321x321xx2

Substitute the solved value into one of the original equations and solve.

1yx

Write the answer as an ordered pair. Remember to place the x value first.

(–1,0)

1y

1x 2yx2 and 1yx

Watch one more time on how to do this problem!

0y

1y1

1

2 ) ( x2

Solve the linear system using the substitution method.

1xy

1x3x321x321xx2

1yx

(–1,0)

1y

1x 2yx2 and 1yx

0y

1y1

1

Solve the linear system using substitution.

3 y2x 2 and 1y4x Example 1Example 2Example 3

15y6x2 and 0y2x

10yx2 and 5yx3

3y2 2

Example 1 Solve the linear system.

1y4x

21

y

5y1032y103y22y8

1x12x

121

4x

21

,1

1y4

1y4x 3 y2x 2 and 1y4x

Example 2 Solve the linear system.

y2x

23

y

1015

y

15y1015y6y4

3x03x

023

2x

0y2x 15y6 2

23

,3

y20y2x 15y6x2 and

10 x2 5yx3

Example 3 Solve the linear system.

5x3y

3x15x5105x5105x3x2

4y5y95y33

(3,–4)

5x3 10yx2 and 5yx3

Did you distribute

the negative correctly?

and7yx3

Solve the linear system.

7x3y

8y2x6

8 2x6

Write in your notes: When solving produces a false statement, there is no solution.

What would the graph of this system look like to show “no solution”?

No solution

7x3

814814x6x68 7x32x6

x3 x3

x

yParallel lines

6y6x3and2y2x

Solve the linear system.

Write in your notes: When solving produces a true statement, there are infinitely many solutions.

What would the graph of this system look like to show “infinitely many solutions”?

Infinitely many solutions

y2 y22y2x

1 1 12y2x

6y63

2y2

6y66y6 6 6

00

x

y

Same lines

Practice Problems. Use substitution to solve each system of equations. If the system does not have exactly one solution, state whether it has no solution or infinitely many solutions. If the system has one solution, name it.

1yx25yx2 .1

6y2x43yx2 .2

4y6x24y3x .6

4y2x3yx .5

4y2x4y2x .4

0y2x44yx2 .3

no solution

Infinitely many solutions

(1,2)

Infinitely many solutions

(2,1)

no solution

5-A3 Pages 263-265 #8–19,43–48.