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Solving Systems of
Equations by
Graphing
Algebra 1 ~ Chapter 7.1
Solving Systems by Graphing
** A system of linear equations is a set of two
or more linear equations containing two or
more variables and connected with a bracket.
** A solution of a system of linear equations
with two variables is an ordered pair that
satisfies each equation in the system.
** So, if an ordered pair is a solution, it will
make both equations true.
Solving
byare
Graphing
All solutions
of aSystems
linear equation
on its graph. To
find a solution of a system of linear equations, you
need a point that each line has in common. In other
words, you need their point of intersection.
y = 2x – 1
y = –x + 5
The point (2, 3) is where the
two lines intersect and is a
solution of both equations, so
(2, 3) is the solution of the
systems.
Solving
Systems
by Graphing
Checking
to make
sure you graphed
the lines correctly,
therefore checking for SURE your answer.
y = 2x – 1
y = –x + 5
In the previous slide we graphed
the 2 lines and found (2, 3) to be
the solution.
Check your answer by plugging in (2, 3) to each line.
y = 2x – 1
y = -x + 5
3 = 2(2) – 1
3 = -(2) + 5
3=4–1
3 = -2 + 5
3=3

3=3 
by Graphing
Ex. 1 Solving
- Solve theSystems
system by graphing,
then check
your solution.
y=x
y = –2x – 3
The solution appears to be at (–1, –1).
y=x
Check
Substitute (–1, –1) into the
system.
y = –2x – 3
y=x
(–1)
•
y = –2x – 3
–1
(–1)
–1

(–1) –2(–1) –3
–1
2–3
–1 – 1 
The solution is (–1, –1).
Graphing
Ex. 2 Solving
- Solve theSystems
system by by
graphing.
Check your
solution.
The solution appears to be (–2, 3).
y = –2x – 1
y=x+5
y=x+5
y = –2x – 1
Check Substitute (–2, 3) into
the system.
y = –2x – 1
y=x+5
3
3
3
3 –2 + 5
3 3
–2(–2) – 1
4 –1
3
The solution is (–2, 3).
Systems
Graphing
Ex. 3Solving
- Solve the
system byby
graphing.
Check
your answer.
2x + y = 4
Rewrite the second equation
in slope-intercept form.
y = –2x + 4
2x + y = 4
–2x
– 2x
y = –2x + 4
The solution appears to be (3, –2).
Solving Systems by Graphing
Example 3 Continued …. CHECK
2x + y = 4
Check Substitute (3, –2) into the
system. Into the ORIGINAL
equations.
2x + y = 4
–2
–2
–2
2(3) + (–2) 4
(3) – 3
6–2 4
4 4
1–3
–2

The solution is (3, –2).
Solving Systems by Graphing
Ex. 4 - Tell whether the ordered pair is a solution
of the given system.
x + 3y = 4 4
(–2, 2);
–x + y = 2
x + 3y = 4
–2 + (3)2 4
–2 + 6
4
4
4
–x + y = 2
Substitute –2
for x and 2
for y.
–(–2) + 2 2
4
2
The ordered pair (–2, 2) makes one equation true, but
not the other. (-2, 2) is NOT a solution of this system.
Systems
by Graphing
Ex. 5 Solving
- Tell whether
the ordered
pair is a solution
of the given system.
(5, 2);
3x – y = 13
3x – y = 13
0
2–2 0
0 0
Substitute 5
for x and 2
for y.
3(5) – 2
15 – 2
13
13
13
13 
The ordered pair (5, 2) makes both equations true, (5, 2)
is the solution of this system.
Solving
Systems by Graphing
Number
of Solutions
bythe
Graphing
Ex. 6 Solving
– Number ofSystems
solutions. Use
graph to
determine whether each system has no solution, one
solution, or infinitely many solutions.
a.) y = -x + 5
One solution
y=x–3
Consistent/independent
b.) y = -x + 5
No solutions
Inconsistent
2x + 2y = -8
c.) 2x + 2y = -8
y = -x - 4
Infinitely many solutions
Consistent/Dependent
Solving
byApplication
Graphing
Ex. 7: Systems
Problem-Solving
Wren and Jenni are reading the same book. Wren is on
page 14 and reads 2 pages every night. Jenni is on page 6
and reads 3 pages every night. After how many nights will
they have read the same number of pages? How many
pages will that be?
Wren
p
=
2
n
+
14
Jenni
p
=
3
n
+
6
SolvingExample
Systems
by Graphing
7 Continued
Graph p = 2n + 14 and p = 3n + 6. The lines appear to intersect
at (8, 30). So, the number of pages read will be the same at
8 nights with a total of 30 pages.
(8, 30)
Nights
SolvingExample
Systems
by Graphing
7 Continued
Check (8, 30) using both equations.
After 8 nights, Wren will have read 30 pages:

2(8) + 14 = 16 + 14 = 30
After 8 nights, Jenni will have read 30 pages:
3(8) + 6 = 24 + 6 = 30

Solving Systems
by
Example
8 Graphing
Video club A charges $10 for membership and $3 per
movie rental. Video club B charges $15 for membership
and $2 per movie rental.
For how many movie rentals will the cost be the same at
both video clubs? What is that cost?
Club A
c
=
3

r
+
10
Club B
c
=
2

r
+
15
SolvingExample
Systems
by Graphing
8 Continued
Graph c = 3r + 10 and c = 2r + 15. The lines appear to
intersect at (5, 25). So, the cost will be the same for 5
rentals and the total cost will be $25.
SolvingExample
Systems
by Graphing
8 Continued
Check (5, 25) using both equations.
Number of movie rentals for Club A to reach $25:
3(5) + 10 = 15 + 10 = 25 
Number of movie rentals for Club B to reach $25:
2(5) + 15 = 10 + 15 = 25 
Lesson Wrap
Solving Systems
by Up
Graphing
Tell whether the ordered pair is a solution of the given
system. Remember you do NOT have to graph the lines
to answer these questions.
1. (–3, 1);
2. (2, –4);
no
yes
Solving Systems by Graphing
Solve and CHECK the system by graphing.
3.
y + 2x = 9
y = 4x – 3
(2, 5)