1. Solving Systems of Linear Equations by Graphing Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. Solving Systems of Linear Equations by Graphing Warm Up Graph each equation. 1. y = x + 2 2. y = x – 3 3. y = 2x + 3 4. y = x 2

3. Solving Systems of Linear Equations by Graphing Problem of the Day Write equations in slope-intercept form for a set of parallel lines. Then write two equations in slope-intercept form for two intersecting lines. Possible answers: parallel: y = 3x + 4 and y = 3x + 2 intersecting: y = x + 5 and y = 2x

4. Solving Systems of Linear Equations by Graphing Learn to graph systems of linear equations to find their solutions.

5. Solving Systems of Linear Equations by Graphing A fishing boat leaves the harbor traveling east at 16 knots (nautical miles per hour). After it travels 40 nautical miles, a Coast Guard cutter follows the boat, traveling at 26 knots. After how many hours will the Coast Guard Cutter catch up with the fishing boat? Additional Example 1: Graphing a System of Linear Equations by Graphing Let t = time in hours Let d = distance in nautical miles Fishing boat distance: d = 16t + 40 Coast Guard cutter distance: d = 26t

6. Solving Systems of Linear Equations by Graphing Additional Example 1 Continued Graph each equation. The point of intersection appears to be (4, 104). Check d = 16t + 40 104 = 16(4) + 40 ? 104 = 104 d = 26t 104 = 26(4) ? 104 = 104 The Coast Guard cutter will catch up after 4 hours, 104 nautical miles from the harbor.

7. Solving Systems of Linear Equations by Graphing A bus leaves the school traveling west at 50 miles per hour. After it travels 15 miles, a car follows the bus, traveling at 55 miles per hour. After how many hours will the car catch up with the bus? Check It Out: Example 1 Let t = time in hours Let d = distance in miles bus distance: d = 50t + 15 car distance: d = 55t

8. Solving Systems of Linear Equations by Graphing Check It Out: Example 1 Continued Graph each equation. The point of intersection appears to be (3, 165). Check d = 50t + 15 165 = 50(3) + 15 ? 165 = 165 d = 55t 165 = 55(3) ? 165 = 165 The car will catch up after 3 hours, 165 miles from the school. 1 2 3 4 5 6 7 8 9 10 50 100 150 200 Time (h) Distance (mi)

9. Solving Systems of Linear Equations by Graphing Not all systems of linear equations have graphs that intersect in one point. There are three possibilities for the graph of a system of two linear equations, and each represents a different solution set.

10. Solving Systems of Linear Equations by Graphing

11. y = 2x – 7 Additional Example 2A: Solving Systems of Linear Equations by Graphing Step 1: Solve both equations for y. 3x + y = 3 y = 2x – 7 3x + y = 3 –3x y = 3 – 3x Step 2: Graph. The lines intersect at (2, –3), so the solution is (2, –3).

12. Solving Systems of Linear Equations by Graphing Additional Example 2A Continued Check y = 2x – 73x + y = 3 ? –3 = 2(2) – 7 ? –3 = –3 3(2) + (–3) = 3 ? 3 = 3 ?

13. Solving Systems of Linear Equations by Graphing 2x + y = 9 Additional Example 2B: Solving Systems of Linear Equations by Graphing Step 1: Solve both equations for y. y – 9 = –2x 2x + y = 9 y – 9 = –2x + 9 +9 y = –2x + 9 Step 2: Graph. The lines are the same, so the system has infinitely many solutions. –2x y = –2x + 9

14. Solving Systems of Linear Equations by Graphing Additional Example 2B Continued Check ? 9 = 9 +2x –2x + 9 = –2x + 9 ? y = y ?

15. Solving Systems of Linear Equations by Graphing y = –4x + 1 Check It Out: Example 2 Step 1: Solve both equations for y. 5x + y = –1 y = –4x + 1 5x + y = –1 –5x y = –5x – 1 Step 2: Graph. The lines are intersect at (–2, 9), so the solution is (–2, 9).

16. Solving Systems of Linear Equations by Graphing Check It Out: Example 2 Continued Check y = –4x + 15x + y = –1 ? 9 = –4(–2) + 1 ? 9 = 9 5(–2) + (9) = –1 ? –1 = –1 ?

17. Solving Systems of Linear Equations by Graphing Standard Lesson Quiz Lesson Quizzes Lesson Quiz for Student Response Systems

18. Solving Systems of Linear Equations by Graphing Solve each system of equations by graphing. Check your answer. 1. A car left Cincinnati traveling 55 mi/h. After it had driven 225 miles, a second car left Cincinnati on the same route traveling 70 mi/h. How long after the 2nd car leaves will it reach the first car? Lesson Quiz 2. y = x; y = 3x 3. y = 4 – x; x + y = 1 15 h (0, 0) no solution

19. Solving Systems of Linear Equations by Graphing 1. Solve the system of equations. y = 2 – x 2y = 4 – 2x A. no solution B. infinitely many solutions C. (1, 1) D. (2, 2) Lesson Quiz for Student Response Systems

20. Solving Systems of Linear Equations by Graphing 2. Solve the system of equations. y = 5 – x 3 – x = y A. no solution B. infinitely many solutions C. (3, 5) D. (5, 3) Lesson Quiz for Student Response Systems

21. Solving Systems of Linear Equations by Graphing 3. Solve the system of equations. y = 5 – 2x 3x = y A. no solution B. infinitely many solutions C. (1, 3) D. (3, 1) Lesson Quiz for Student Response Systems