1. The document provides examples for writing and solving systems of linear equations by graphing. It explains how to write equations in slope-intercept form, graph the lines on the same coordinate plane, and determine if the system has no solution, one solution, or an infinite number of solutions based on where the lines intersect. 2. Key steps shown include writing equations in y=mx+b form, finding the slope and y-intercept, graphing the lines and analyzing where they intersect to determine the number of solutions. 3. Graphing systems of equations allows you to visually see where lines intersect, satisfying both equations simultaneously and showing the solution to the system.

1. Course 3, Lesson 3-7
Write an equation in point-slope form and slope-intercept form
for each line.
1. passes through (2, 5), slope = –1
2. passes through (1, 9), slope = 4
3. passes through (3, 6) and (1, 2)
4. Bode is planning a party. The cost for 10 people is $170. The cost
for 25 people is $470. Write an equation in point-slope form to
represent the cost y of having a party for x people.
5. Write the equation in point-slope form that represents the table of
values shown.

2. Course 3, Lesson 3-7
ANSWERS
1. y – 5 = –1(x – 5); y = –x + 7
2. y – 9 = 4(x – 1); y = 4x + 5
3. y – 6 = 2(x – 3); y = 2x
4. Sample answer: y – 170 = 20(x – 10)
5. y – 3 = 2(x – 1)

3. WHY are graphs helpful?
Expressions and Equations
Course 3, Lesson 3-7

4. • 8.EE.8
Analyze and solve pairs of simultaneous linear equations.
• 8.EE.8a
Understand that solutions to a system of two linear equations in two
variables correspond to points of intersection of their graphs, because
points of intersection satisfy both equations simultaneously.
• 8.EE.8b
Solve systems of two linear equations in two variables algebraically, and
estimate solutions by graphing the equations. Solve simple cases by
inspection.
Course 3, Lesson 3-7 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
Expressions and Equations

5. • 8.EE.8c
Solve real-world and mathematical problems leading to two linear
equations in two variables.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
7 Look for and make use of structure.
Course 3, Lesson 3-7 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of
Chief State School Officers. All rights reserved.
Expressions and Equations

6. To
• write and solve a system of equations by
graphing
• determine if the system has no solution,
one solution, or an infinite number of
solutions
Course 3, Lesson 3-7
Expressions and Equations

7. • systems of equations
Course 3, Lesson 3-7
Expressions and Equations

8. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
1. Solve the system y = –2x – 3 and y = 2x + 5 by graphing.
Graph each equation on the same
coordinate plane.
The graphs appear to intersect at (–2, 1).
Check
7
Check this estimate by replacing
x with –2 and y with 1.
y = 2x + 5y = –2x – 3
1 = 2(–2) + 51 = –2(–2) – 3
? ?
1 = 1 1 = 1
The solution of the system is (–2, 1).

9. Answer
Need Another Example?
Solve the system y = 3x – 2 and y = x + 1 by
graphing.
(1.5, 2.5)

10. 1
Need Another Example?
2
3
Step-by-Step Example
2. Gregory’s Motorsports has motorcycles (two wheels) and
ATVs (four wheels) in stock. The store has a total of 45
vehicles, that, together, have 130 wheels.
Let y represent the motorcycles and x represent the ATVs.
y + x = 45
Write a system of equations that represents the situation.
The number of motorcycles and ATVs is 45.
2y + 4x = 130 The number of wheels equals 130.

11. Answer
Need Another Example?
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.
x + y = 14; 6x + 4y = 72

12. 1
Need Another Example?
2
3
4
Step-by-Step Example
3. GreGregory’s Motorsports has motorcycles (two wheels) and ATVs
(four wheels) in stock. The store has a total of 45 vehicles, that,
together, have 130 wheels. The situation can be represented by the
equations x + y = 45 and 2y + 4x = 130.
Solve the system of equations. Interpret the solution.
Write each equation in slope-intercept form.
x + y = 45
Graph both equations on the same coordinate
plane. The equations intersect at (20, 25).
2y + 4x = 130
y = –x + 45 2y = –4x + 130
y = –2x + 65
The solution is (20, 25). This means that the
store has 20 ATVs and 25 motorcycles.
x + y = 45
20 + 25 = 45
45 = 45
2y + 4x = 130
2(25) + 4(20) = 130
130 = 130
Check
? ?

13. Answer
Need Another Example?
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 situation can be represented
by x + y = 14 and 6x + 4y = 72. Solve the system of equations.
Interpret the solution.
(8, 6); Ms. Baker bought 8 packages of
red pens and 6 packages of green pens.

14. 1
Need Another Example?
2
3
4
Step-by-Step Example
4. Solve the system of equations by graphing.
y = 2x + 1
y = 2x – 3
Graph each equation on
the same coordinate plane.
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.
Since y – 2x cannot simultaneously be 1 and –3, there is no solution.
Check Analyze the equations. Write them in standard form.
y = 2x + 1
y – 2x = 2x – 2x + 1
y – 2x = 1
y = 2x – 3
y – 2x = 2x – 2x – 3
y – 2x = –3

15. Answer
Need Another Example?
Solve the system y = x – 1 and y = x by graphing.
no solution

16. 1
Need Another Example?
2
3
Step-by-Step Example
5. Solve the system of equations by graphing.
y = 2x + 1
y – 3 = 2x – 2
Write y – 3 = 2x – 2 in slope-intercept form.
Both equations are the same.
Graph the line.
Any ordered pair on the graph
will satisfy both equations. So,
there are an infinite number of
solutions of the system.
y – 3 = 2x – 2
y – 3 + 3 = 2x – 2 + 3
y = 2x + 1
Write the equation.
Add 3 to each side.
Simplify.

17. Answer
Need Another Example?
Solve the system y = 3x – 2 and y – 2x = x – 2
by graphing.
an infinite number of solutions

18. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
6. A system of equations consists of two lines. One line passes through (2, 3) and
(0, 5). The other line passes through (1, 1) and (0, –1). Determine if the system
has no solution, one solution, or an infinite number of solutions.
To compare the two lines, write the equation of each line in slope-intercept form.
Find the slope of each line.
y = mx + b
y = –1x + 5
(2, 3) and (0, 5) (1, 1) and (0, –1)
Find the y-intercept for each line. Then write the equation.
Use the point (0, 5).
The y-intercept is 5.
Use the point (0, –1).
The y-intercept is –1.
y = mx + b
y = 2x – 1
Since the lines have different slopes and
different y-intercepts, they intersect in
exactly one point.
Check
The lines intersect at (2, 3) so there is
exactly one solution.
Graph each line on a coordinate plane.

19. Answer
Need Another Example?
A system of equations consists of two lines. One
line passes through (–3, 9) and (2, 6). The other
line passes through (–5, 7) and (2, 14). Determine
if the system has no solution, one solution, or an
infinite number of solutions.
one solution

20. How did what you learned
today help you answer the
WHY are graphs helpful?
Course 3, Lesson 3-7
Expressions and Equations

21. How did what you learned
today help you answer the
WHY are graphs helpful?
Course 3, Lesson 3-7
Expressions and Equations
Sample answers:
• You can graph a system of equations to find the point
of intersection of the lines which is the solution.
• You can tell if two lines are parallel which means the
system has no solutions.
• You can tell when two lines are the same line which
means the system has an infinite number of solutions.

22. Solve the system y = 2x + 6
and y = – 4x – 6 by
graphing.
Ratios and Proportional RelationshipsExpressions and Equations
Course 3, Lesson 3-7