Warm Up Problem of the Day Lesson Presentation Lesson Quizzes
Warm Up 1. y = x + 2 2. y = x – 3 3. y = 2x + 3 4. y = x2 Graph each equation. 1. y = x + 2 2. y = x – 3 3. y = 2x + 3 4. y = x2
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
Learn to graph systems of linear equations to find their solutions.
Additional Example 1: Graphing a System 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? Let t = time in hours Let d = distance in nautical miles Fishing boat distance: d = 16t + 40 Coast Guard cutter distance: d = 26t
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.
Check It Out: Example 1 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? Let t = time in hours Let d = distance in miles bus distance: d = 50t + 15 car distance: d = 55t
Check It Out: Example 1 Continued Graph each equation. The point of intersection appears to be (3, 165). Check d = 50t + 15 200 165 = 50(3) + 15 ? 150 Distance (mi) 165 = 165 100 50 d = 55t 165 = 55(3) ? 1 2 3 4 5 6 7 8 9 10 Time (h) 165 = 165 The car will catch up after 3 hours, 165 miles from the school.
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.
Additional Example 2A: Solving Systems of Linear Equations by Graphing y = 2x – 7 3x + y = 3 Step 1: Solve both equations for y. 3x + y = 3 y = 2x – 7 –3x –3x y = 3 – 3x Step 2: Graph. The lines intersect at (2, –3), so the solution is (2, –3).
Additional Example 2A Continued Check y = 2x – 7 3x + y = 3 ? –3 = 2(2) – 7 3(2) + (–3) = 3 ? ? –3 = –3 3 = 3 ?
Additional Example 2B: Solving Systems of Linear Equations by Graphing 2x + y = 9 y – 9 = –2x Step 1: Solve both equations for y. 2x + y = 9 y – 9 = –2x –2x –2x + 9 +9 y = –2x + 9 y = –2x + 9 Step 2: Graph. The lines are the same, so the system has infinitely many solutions.
Additional Example 2B Continued Check y = y ? –2x + 9 = –2x + 9 ? +2x +2x ? 9 = 9
Check It Out: Example 2 y = –4x + 1 5x + y = –1 Step 1: Solve both equations for y. y = –4x + 1 5x + y = –1 –5x –5x y = –5x – 1 Step 2: Graph. The lines are intersect at (–2, 9), so the solution is (–2, 9).
Check It Out: Example 2 Continued y = –4x + 1 5x + y = –1 ? 9 = –4(–2) + 1 5(–2) + (9) = –1 ? ? 9 = 9 –1 = –1 ?
Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 17 17
Lesson Quiz 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? 15 h 2. y = x; y = 3x (0, 0) 3. y = 4 – x; x + y = 1 no solution
Lesson Quiz for Student Response Systems 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) 19 19
Lesson Quiz for Student Response Systems 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) 20 20
Lesson Quiz for Student Response Systems 3. Solve the system of equations. y = 5 – 2x 3x = y A. no solution B. infinitely many solutions C. (1, 3) D. (3, 1) 21 21