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A system of equations is a set of two or more equations that are to be solved.

A system of equations is a set of two or more equations that are to be solved. A solution of a system of two equations in two variables is an ordered pair of numbers that makes both equations true. A solution to two equations (1, 3) for y = 2x + 1 and y = 5x - 2. –3 x –3 x.

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A system of equations is a set of two or more equations that are to be solved.

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  1. A system of equations is a set of two or more equations that are to be solved.

  2. A solutionof a system of two equations in two variables is an ordered pair of numbers that makes both equations true. A solution to two equations (1, 3) for y = 2x + 1 and y = 5x - 2

  3. –3x –3x Example 1: 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 y = 3 – 3x Step 2: Graph. The lines intersect at (2, –3), so the solution is (2, –3).

  4. ? ? –3 = 2(2)– 7 3(2) + (–3) = 3 ? ? –3 = –3  3 = 3  Example 1A Continued Check y = 2x – 7 3x + y = 3

  5. 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.

  6. + 9+9 –2x –2x Example 2: 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 y = –2x + 9 y = –2x + 9 Step 2: Graph. The lines are the same, so the system has infinitely many solutions.

  7. ? y = y ? –2x + 9 = –2x + 9 +2x +2x ? 9 = 9  Example 1B Continued Check

  8. –5x –5x Example 3 y = –4x + 1 5x + y = –1 Step 1: Solve both equations for y. y = –4x + 1 5x + y = –1 y = –5x – 1 Step 2: Graph. The lines are intersect at (–2, 9), so the solution is (–2, 9).

  9. ? ? 9 = –4(–2)+ 1 5(–2) + (9) = –1 ? ? 9 = 9  –1 = –1  Example 3 Continued Check y = –4x + 1 5x + y = –1

  10. Application: 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

  11. ? 165 = 50(3) + 15 ? 165 = 55(3) Application: Example 1 Continued Graph each equation. The point of intersection appears to be (3, 165). Check d = 50t + 15 200 150 Distance (mi) 100 165 = 165  50 d = 55t 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.

  12. 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

  13. 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)

  14. 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)

  15. 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)

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