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 Lesson Menu

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

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

Answer: The solution of the system is (1.5, 2.5). 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. Check y = 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

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

x + y = 14 The number of packages of pens is 14. 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

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 = 23 x + y = 5 B. x + y = 23 x + y = 9 C. 2x + 3y = 23 x + y = 9 D. 2x + 3y = 9 x + y = 23 Example 2 CYP

Write each equation in slope-intercept form. x + y = 14 6x + 4y = 72 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 y = – x + 18 Example 3

Choose values for x that could satisfy the equations. 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

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

A. (4, 5); They bought 4 tacos and 5 burritos. 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

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

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

Solve the system y = –x – 4 and x + y = 1 by graphing. C. no solution D. infinitely many solutions Example 4 CYP

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

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

Solve the system y = – x + 2 and 3x + 2y = 2 by graphing. C. no solution D. infinitely many solutions Example 5 CYP