2. 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

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

4. 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

5. 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) – = (1.5) + 1
2.5 = 2.5  2.5 = 2.5 
Answer: The solution of the system is (1.5, 2.5).
Example 1

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

7. 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

8. 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

9. 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 = x + 4y = 72
y = –x y = –6x + 72
y = – x + 18
Example 3

10. 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

11. 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

12. 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

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

14. 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

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

16. 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

17. 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

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