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Solve Systems of Equations by Graphing

Learn how to solve systems of equations by graphing. Determine the number of solutions based on the intersection points on the graph. Practice real-world examples.

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Solve Systems of Equations by Graphing

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  1. Lesson Menu Main Idea and New Vocabulary Example 1: One Solution Example 2: Real-World Example Example 3: Real-World Example Example 4: No Solution Example 5: Infinitely Many Solutions

  2. Solve systems of equations by graphing. • system of equations Main Idea/Vocabulary

  3. One Solution Solve the system y = 3x – 2 and y = x + 1 by graphing. Graph each equation on the same coordinate plane. Example 1

  4. One Solution The graphs appear to intersect at (1.5, 2.5). Check this estimate by replacing x with 1.5 and y with 2.5. Checky= 3x– 2 y= x+ 1 2.5= 3(1.5)– 2 2.5= (1.5)+ 1 2.5 = 2.5  2.5 = 2.5  Answer: The solution of the system is (1.5, 2.5). Example 1

  5. Solve the system y = – x + 3 and y = –x + 2 by graphing. A. (2, 0) B. (0, 3) C. (–2, 2) D. (–2, 4) Example 1 CYP

  6. One Solution PENS Ms.Baker bought 14 packages of red and green pens for a total of 72 pens. The red pens come in packages of 6 and the green pens come in packages of 4. Write a system of equations that represents the situation. Let x represent packages of red pens and y represent packages of green pens. x + y = 14 The number of packages of pens is 14. 6x + 4y = 72 The number of pens equals 72. Answer: The system of equations is x + y = 14 and 6x + 4y = 72. Example 2

  7. FOOD Abigail and her friends bought some tacos and burritos and spent $23. The tacos cost $2 each and the burritos cost $3 each. They bought a total of 9 items. Write a system of equations that represents the situation. A.2x + 3y = 23x + y = 5 B.x + y = 23x + y = 9 C.2x + 3y = 23x + y = 9 D.2x + 3y = 9x + y = 23 Example 2 CYP

  8. y = –x + 18 One Solution PENS Ms.Baker bought 14 packages of red and green pens for a total of 72 pens. The red pens come in packages of 6 and the green pens come in packages of 4. The system of equations that represents the situation is x + y = 14 and 6x + 4y = 72. Solve the system of equations. Interpret the solution. Write each equation in slope-intercept form. x + y = 14 6x + 4y = 72 y = –x + 14 4y = –6x + 72 Example 3

  9. One Solution Choose values for x that could satisfy the equations. Both equations have the same value when x = 8 and y = 6. You can also graph both equations on the same coordinate plane. Example 3

  10. One Solution Answer: The equations intersect at (8, 6). The solution is (8, 6). This means that Ms. Baker bought 8 packages of red pens and 6 packages of green pens. Example 3

  11. FOOD Abigail and her friends bought some tacos and burritos and spent $23. The tacos cost $2 each and the burritos cost $3 each. They bought a total of 9 items. The system of equations that represents the situation is x + y = 9 and 2x + 3y = 23. Solve the system of equations. Interpret the solution. A.(4, 5); They bought 4 tacos and 5 burritos. B.(4, 5); They bought 4 burritos and 5 tacos. C.(5, 4); They bought 5 tacos and 4 burritos. D.(6, 3); They bought 6 tacos and 3 burritos. Example 3 CYP

  12. No Solution Solve the system y = 2x – 1 and y = 2x by graphing. Graph each equation on the same coordinate plane. Example 4

  13. No Solution The graphs appear to be parallel lines. Since there is no coordinate point that is a solution of both equations, there is no solution for this system of equations. Answer: no solution Example 4

  14. Solve the system y = –x – 4 and x + y = 1by graphing. A. (2.5, –1.5) B. (–2.5, –6.5) C. no solution D. infinitely many solutions Example 4 CYP

  15. Infinitely Many Solutions Solve the system y = 3x − 2 and y − 2x = x − 2 by graphing. Write y − 2x = x − 2 in slope-intercept form. y – 2x = x – 2 Write the equation. y – 2x + 2x = x – 2 + 2x Add 2x to each side. y = 3x – 2 Simplify. Both equations are the same. Graph the equation. Example 5

  16. Infinitely Many Solutions Any ordered pair on the graph will satisfy both equations. So, there are infinitely many solutions of the system. Answer: infinitely many solutions Example 5

  17. Solve the system y = – x + 2 and 3x + 2y = 2 by graphing. A. (0, 2) B. (2, –1) C. no solution D. infinitely many solutions Example 5 CYP

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