Solving Systems of Linear Equations by
12-7 Graphing
A system of equations is
a set of two or more
equations that are to be
solved.
Solving Systems of Linear Equations by
12-7 Graphing
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
Solving Systems of Linear Equations by
12-7 Graphing
Example 1: Solving Systems of Linear Equations by
Graphing
y = 2x – 7
3x + y = 3
Step 1: Solve both equations for y.
y = 2x – 7
3x + y = 3
–3x
–3x
y = 3 – 3x
Step 2: Graph.
The lines intersect at (2, –3),
so the solution is (2, –3).
Solving Systems of Linear Equations by
12-7 Graphing
Example 1A Continued
Check
y = 2x – 7
?
–3 = 2(2) – 7
?
–3 = –3 
3x + y = 3
?
3(2) + (–3) = 3
?
3=3
Solving Systems of Linear Equations by
12-7 Graphing
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.
Solving Systems of Linear Equations by
12-7 Graphing
Solving Systems of Linear Equations by
12-7 Graphing
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
–2x
–2x
y = –2x + 9
y – 9 = –2x
+9
+9
y = –2x + 9
Step 2: Graph.
The lines are the same,
so the system has
infinitely many solutions.
Solving Systems of Linear Equations by
12-7 Graphing
Example 1B Continued
Check
?
y=y
?
–2x + 9 = –2x + 9
+2x
+2x
?
9=9
Solving Systems of Linear Equations by
12-7 Graphing
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).
Solving Systems of Linear Equations by
12-7 Graphing
Example 3 Continued
Check
y = –4x + 1
?
9 = –4(–2) + 1
?
9=9
5x + y = –1
?
5(–2) + (9) = –1
?
–1 = –1 
Solving Systems of Linear Equations by
12-7 Graphing
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
Solving Systems of Linear Equations by
12-7 Graphing
Application: Example 1 Continued
Graph each equation. The point of
intersection appears to be (3, 165).
d = 50t + 15
?
165 = 50(3) + 15
165 = 165 
d = 55t
?
165 = 55(3)
165 = 165 
Distance (mi)
Check
200
150
100
50
1 2 3 4 5 6 7 8 9 10
Time (h)
The car will catch up after 3 hours, 165 miles from
the school.
Solving Systems of Linear Equations by
12-7 Graphing
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
Solving Systems of Linear Equations by
12-7 Graphing
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)
Solving Systems of Linear Equations by
12-7 Graphing
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)
Solving Systems of Linear Equations by
12-7 Graphing
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)