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Main Idea and New Vocabulary
NGSSS
Example 1: One Solution
Example 2: Real-World Example
Example 3: Real-World Example
Example 4: No Solution
Example 5: Infinitely Many Solutions
Five-Minute Check
• Solve systems of equations by graphing.
• system of equations
MA.8.A.1.3 Use tables, graphs, and models
to represent, analyze, and solve real-world
problems related to systems of linear
equations.
MA.8.A.1.4 Identify the solution to a system
of linear equations using graphs.
One Solution
Solve the system y = 3x – 2 and y = x + 1 by
graphing.
Graph each equation on the same coordinate plane.
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).
Solve the system y = –
graphing.
A. (2, 0)
B. (0, 3)
C. (–2, 2)
D. (–2, 4)
x + 3 and y = –x + 2 by
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
6x + 4y = 72
The number of packages of pens is 14.
The number of pens equals 72.
Answer: The system of equations is x + y = 14 and
6x + 4y = 72.
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
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
y = –x + 14
6x + 4y = 72
4y = –6x + 72
y = – x + 18
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.
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.
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.
No Solution
Solve the system y = 2x – 1 and y = 2x by
graphing. Graph each equation on the same
coordinate plane.
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
Solve the system y = –x – 4 and x + y = 1 by
graphing.
A. (2.5, –1.5)
B. (–2.5, –6.5)
C. no solution
D. infinitely many solutions
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
y – 2x + 2x = x – 2 + 2x
y = 3x – 2
Write the equation.
Add 2x to each side.
Simplify.
Both equations are the same. Graph the equation.
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
Solve the system y = – x + 2 and 3x + 2y = 2 by
graphing.
A. (0, 2)
B. (2, –1)
C. no solution
D. infinitely many solutions
Last year, Justin and his sister, Karin, earned a
total of $468 in allowance. If Justin earned $52
more than Karin in allowance, write a system of
equations that represents their allowances.
A. j + k = 468
j = 52 + k
C.
j + k = 468
j = 52 – k
B. j + k = 468
k = 52 + j
D.
j – k = 468
j = 52 + k
Mrs. Kung spent the same amount on two
programs at the local recreation center. The
aerobics class costs an initial fee of $10 plus $3
per class. The pottery class costs an initial fee of
$6 plus $5 per class. Write a system of equations
to represent the cost for the two programs.
A. 10x + 3x = y
6x + 5x = y
C. 10 + 3x = y
6 + 5x = y
B. 10x + 3 = y
6x + 5 = y
D. –10 + 3x = y
–6 + 5x = y
The sum of Dewan’s age and three times Adrianne’s
age is 32. The difference between Dewan’s age and
Adrianne’s age is 4. Which system can be used to
find Dewan’s age and Adrianne’s age?
A. d + 3a = 32
3d − a = 4
B. 3d + a = 32
a−d=4
C. d + 3a = 32
d−a=4
D. d + 3 + a = 32
d−a=4