Over Lesson 8–9

A. A

B. B

C. C

D. D

Which is an equation for the line of best fit for the scatter plot where x is the years since 1998 and y is the number of visitors in thousands?

A. y = 3x + 15B. y = x + 5C. y = 3x – 15D.

Use the line of best fit from above to predict the number of visitors in 2010.

You have already solved linear equations by graphing. (Lesson 8–2)

• Solve systems of linear equations by graphing.

• Solve systems of linear equations by substitution.

• system of equations

• substitution

A set of two or more equations with the same variables

Use algebraic methods to find an exact solution of a system of equations.

Solve by Graphing

Solve the system of equations by graphing.

y = –x + 4y = 2x + 1

Graph each line.

Solve by Graphing

Answer: The solution of the system is (1, 3).

The graphs appear to intersect at (1, 3). Check this by replacing x with 1 and y with 3.

Check

y = –x + 4 y = 2x + 1

3 = 3 3 = 33 = –1 + 4 3 = 2(1) + 1

??

A. A

B. B

C. C

D. D

A. (2, 1)

B. (2, –1)

C. (1, 2)

D. (1, –2)

What is the solution of the system of equations?y = –x + 3y = 4x – 2

A. Catori and Mark each download songs. Mark downloaded 4 times as many songs as Catori. Mark also downloaded 6 more songs than Catori.

Write a system of equations to represent this situation.

Let x represent Catori’s songs and y represent Mark’s songs.

y = 4x Mark downloaded 4 times asmany songs as Catori.

y = x + 6 Mark downloaded 6 more songsthan Catori.

Answer: y = 4x and y = x + 6

B. Catori and Mark each download songs. Mark downloaded 4 times as many songs as Catori. Mark also downloaded 6 more songs than Catori.

Solve the system by graphing. Explain what the solution means.

Graph the equations y = 4x and y = x + 6 on the same coordinate grid. The equations intersect at (2, 8).

Answer: So, the solution to the system is x = 2 and y = 8. This means that Catori downloaded 2 songs and Mark downloaded 8 songs.

A. A

B. B

C. C

D. D

A. y = 2xy = x – 8

B. y = 8xy = x – 2

C. y = 2xy = x + 8

D. y = 8xy = x + 2

A. Andy ran twice as far as Ron and Andy ran 8 miles farther than Ron. Which system of equations represents this situation?

A. A

B. B

C. C

D. D

A. Andy ran 10 miles.Ron ran 5 miles.

B. Andy ran 12 miles.Ron ran 6 miles.

C. Andy ran 16 miles.Ron ran 8 miles.

D. Andy ran 18 miles.Ron ran 9 miles.

B. Andy ran twice as far as Ron and Andy ran 8 miles farther than Ron. How far did each person run?

No Solution and Infinitely Many Solutions

A. Solve the system of equations by graphing.

y = –x – 32x + 2y = –6

Both equations have the same graph. Any ordered pair on the graph will satisfy both equations.

Answer: Therefore, there are infinitely many solutions of this system of equations.

No Solution and Infinitely Many Solutions

B. Solve the system of equations by graphing.

y = 2xy = 2x – 4

The graphs appear to be parallel lines.

Answer: Since there is no coordinate pair that is a solution to both equations, there is no solution of this system of equations.

A. A

B. B

C. C

D. D

A. (0, 1)

B. (1, 2)

C. infinitely many solutions

D. no solution

A. Solve the system of equations by graphing.y = x + 12y – 2x = 2

A. A

B. B

C. C

D. D

A. (1, 1)

B. (1, 2)

C. infinitely many solutions

D. no solution

B. Solve the system of equations by graphing.y = xy = x + 1

Solve by Substitution

Solve the system of equations by substitution.

y = 7y = 2x – 5

Replace y with 7 in the second equation.

y = 2x – 5 Write the second equation.

7 = 2x – 5 Replace y with 7.

12 = 2x Add 5 to each side.

6 = x Solve for x.

Answer: The solution of this system of equationsis (6, 7).

A. A

B. B

C. C

D. D

A. (4, 8)

B. (–4, 0)

C. (8, 4)

D. (2, 8)

Solve the system of equations by substitution.

y = 8y = x + 4