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

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

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

7. 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
–2x –2x
y = –2x + 9
y = –2x + 9
Step 2: Graph.
The lines are the same, so the system has infinitely many solutions.

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

9. Example 3
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).

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

11. 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

12. Application: 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) ?
Time (h)
165 = 165 
The car will catch up after 3 hours, 165 miles from the school.

13. 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

14. 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 14

15. 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 15

16. 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) 16 16