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Five-Minute Check (over Lesson 8–9) Then/Now New Vocabulary Example 1: Solve by Graphing Example 2: Standardized Test Example Example 3: No Solution and Infinitely Many Solutions Example 4: Solve by Substitution Concept Summary: Systems of Equations Lesson Menu

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 + 15 B. y = x + 5 C. y = 3x – 15 D. 5-Minute Check 1

Use the line of best fit, y = 3x + 15, to predict the number of visitors in 2010. A. 100,000 B. 80,000 C. 51,000 D. 21,000 5-Minute Check 2

Caleb drew a scatter plot for the average number of hours of homework y he received each year in school x. A line of best fit passed through the points (5, 1.5) and (8, 3.75). Predict the numbers of hours of homework that Caleb can expect to receive in 12th grade. A. 4.75 B. 5.25 C. 6.75 D. 7.25 5-Minute Check 3

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. Then/Now

system of equations substitution Vocabulary

Solve the system of equations by graphing. y = –x + 4 y = 2x + 1 Solve by Graphing Solve the system of equations by graphing. y = –x + 4 y = 2x + 1 Graph each line. Example 1

Answer: The solution of the system is (1, 3). Solve by Graphing 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 = –1 + 4 3 = 2(1) + 1 ? 1 = 1 3 = 3   Answer: The solution of the system is (1, 3). Example 1

What is the solution of the system of equations? y = –x + 3 y = 4x – 2 B. (2, –1) C. (1, 2) D. (1, –2) Example 1

Write a system of equations to represent this situation. 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 as many songs as Catori. y = x + 6 Mark downloaded 6 more songs than Catori. Answer: y = 4x and y = x + 6 Example 2A

Solve the system by graphing. Explain what the solution means. 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. Example 2B

A. Andy ran twice as far as Ron and Andy ran 8 miles farther than Ron A. Andy ran twice as far as Ron and Andy ran 8 miles farther than Ron. Which system of equations represents this situation? A. y = 2x y = x – 8 B. y = 8x y = x – 2 C. y = 2x y = x + 8 D. y = 8x y = x + 2 Example 2 CYP A

A. Andy ran 10 miles. Ron ran 5 miles. B. Andy ran twice as far as Ron and Andy ran 8 miles farther than Ron. How far did each person run? 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. Example 2 CYP B

A. Solve the system of equations by graphing. y = –x – 3 2x + 2y = –6 No Solution and Infinitely Many Solutions A. Solve the system of equations by graphing. y = –x – 3 2x + 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. Example 3A

B. Solve the system of equations by graphing. y = 2x y = 2x – 4 No Solution and Infinitely Many Solutions B. Solve the system of equations by graphing. y = 2x y = 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. Example 3B

A. Solve the system of equations by graphing. y = x + 1 2y – 2x = 2 C. infinitely many solutions D. no solution Example 3A

B. Solve the system of equations by graphing. y = x y = x + 1 C. infinitely many solutions D. no solution Example 3B

Solve the system of equations by substitution. y = 7 y = 2x – 5 Solve by Substitution Solve the system of equations by substitution. y = 7 y = 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 equations is (6, 7). Example 4

Solve the system of equations by substitution. y = 8 y = x + 4 C. (8, 4) D. (2, 8) Example 4

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