Lesson 4: Solving and Graphing Linear Equations

Selected Content Standards

Benchmarks Addressed:
A-2-M Modeling and developing methods for solving equations and
inequalities (e.g., using charts, graphs, manipulatives, and/or
standard algebraic procedures)
A-2-H Recognizing the relationship between operations involving real
numbers and operations involving algebraic expressions.
A-4-H Solving algebraic equations and inequalities using a variety of
techniques with the appropriate tools (e.g., hand-held manipulatives,
graphing calculator, symbolic manipulator, or pencil and paper)

GLEs Addressed:
Grade 8
12. Solve and graph solutions of multi-step linear equations and
inequalities (A-2-M)
Grade 9
14. Graph and interpret linear inequalities in one or two variables and
systems of linear inequalities (A-2-H) (A-4-H)
16. Interpret and solve systems of linear equations using graphing,
substitution, elimination, with and without technology, and matrices
using technology (A-4-H)

Lesson Focus
This lesson is intended to help students see the connections between algebraic
equations and linear graphs. It should include all of the following:
• Graphing algebraic equations with two variables using points on the line
• Writing linear equations from problem situations
• Determining scale for x and y axis
• Interpreting linear graphs and relating them to problem situations

GEE 21 Connection
The skills that will be addressed in this lesson include the following:
• Solve and graph linear equations
• Solve and graph real world problems (addition, subtraction,
multiplication, division) involving linear equations and systems of linear
equations
• Solve and graph multi-step equations

Translating Content Standards into Instruction

A. The first thing we want students to comprehend is the connection between

the algebraic equation and its linear graph. Start the lesson by writing the
simple equation x + y = 10 from Teacher Blackline #1. Discuss how the
solutions would look for this equation. Could x = 4? If x = 4, what would y
Focused Learning Lessons for Mathematics 36 Algebra
 have to be to make the equation true? That is one ordered pair (4,6) that
would be a solution. Are there any more solutions? What about (6,4)? Is
that a different solution? Could x = 0? What would y have to be? Could x be
a fraction? a negative number? a decimal? Get the students to come up
with several examples. In the discussion with the class, be sure to ask the
students to decide what the value of x depends on. Help the students arrive
at the conclusion that the value of y depends upon the value of x. Now ask
the students to graph all of the solutions to the equation, x + y = 10. Do
they notice any pattern? Are there any other solutions that have not been
listed? Where would those solutions go on the graph? Put a few more
solutions on the graph. Could we possibly list all the solutions to the
equation, x + y = 10? How could we show all solutions on the graph? If we
connect all the points in a straight line and put arrows on both ends, then
we can represent all possible solutions to the equation. Name another
solution to the equation and ask the students if it falls on the line. It is
important for the students to know that all points on the line are solutions to
the equation, and all solutions to the equation lie on the line.

Calculator Note: The teacher should lead the students

through graphing this line on their graphing calculators.
Since all linear equations graphed on the calculator must be
in the form of “y =”, the teacher should lead the students
through solving the equation, x + y = 10, for y. The students
should enter the equation into the calculator, graph the
equation, and use the trace function to determine the values
of the points on the line.

B) The students should work to make connections between a problem situation,

the equation that represents the situation, and its graph using example B on
Teacher Blackline # 1: In order to play golf at the Scottsville City Golf
Course, a person must first join the club for a one-time membership fee of
$125 and then pay a $12 green fee each time he plays a round of golf. Ask
the students to decide the total cost of joining the club but not play a round
of golf. Ask the students to determine the total cost of playing one round of
golf. What would be the total cost of 6 rounds of golf? Discuss with the
students what quantities are changing (varying). Since the total cost and the
number of rounds of golf are changing, those two quantities would be our
variables. Discuss with the students what quantity should be the x variable
and what quantity should be the y variable. Remember y depends upon x.
After the students understand that the total cost depends upon the number
of rounds of golf played, then we can assign variables.
x = the number of rounds of golf
y = the total cost of playing golf

Ask the students to come up with an equation to represent the situation.

(y = 125 + 12x)

Discuss with the students the best way to graph the situation. Should they
use the values they have already come up with or should they use the
Focused Learning Lessons for Mathematics 37 Algebra
 equation? How could they use the equation to come up with some ordered
pairs to graph?

Before the students begin to graph the equation, they will need to decide
upon the scale for the x-axis and the scale for the y-axis. Discuss whether or
not they need to be the same scale. Make sure the students understand that
the scale does not need to be 1 unit in length. The y quantities start at 125
and increase by 12. The x quantities start at 0 and increase by 1. After the
scales for the x-axis and the y-axis are decided upon, make sure that the
students label the x-axis with the proper title (number of rounds of golf) and
the y-axis with the proper title (total cost per round). Discuss with the
students what part of the coordinate system will they need. All four
quadrants? The negative sides of the axis? Why aren’t these needed in this
situation?

After the students have decided on the scales and have properly labeled the
x- and y- axes, they should decide how to graph the line. Two points are
needed, but three are best because the third point serves as a double check
of the line. After the points are graphed, they should be connected in a line.

The teacher should lead the students through a discussion about the
connections between the situation, the graph, and the equation. Some
questions that should be raised are
• Does the line go through the origin? Why not?
• Where does the line cross the y-axis? What is the meaning of that point
on the graph in the actual situation? How is that point related to the
equation?
• Does the line cross the x-axis? Why not?
• What is the slope of the line?
• How can you tell from the graph that a round of golf (without the
membership fee) is $12?
• Should the points on the graph really be connected? Does that accurately
represent the situation?

Calculator Note: Get the students to graph the equation on their

graphing calculators. Show the students how to adjust their
window, if needed. The calculator will graph the negative values
for this equation, but we will ignore those points. Show the
students how to access the table on the graphing calculator that
corresponds to the graph.

C. Next we want to make sure that the students can analyze a graph and
describe it using an equation or a real situation.

Have the students work with a partner and analyze the graph on Teacher
Blackline #2. Have the partners discuss the graph and write down as much
information as they can gather from the graph. The partners should assist
each other in raising questions about the graph. After giving the partners

Focused Learning Lessons for Mathematics 38 Algebra

 some time to analyze, the teacher should lead a class discussion on what
information was discovered. The teacher should be sure that information
gathered is supported by what physically appears on the graph. Do not
allow the students to make statements without saying why.

After a complete analysis of the graph, the teacher should discuss with the
students how to write the equation that goes with the graph. The simplest
way for the students to do this might be to identify the y-intercept and the
slope of the line. With the slope and y-intercept identified, the students can
use y = mx + b to write an equation.

Sources of Evidence about Student Learning

A. As students work through the examples, the teacher should monitor their
discussions as they work between the three different representations of a
problem situation, the word problem, the equation, and the graph of that
situation.

B. Have students do the Student Worksheet provided with the lesson.

Focused Learning Lessons for Mathematics 39 Algebra

GEE 21 Connection

Sample items similar to what students might see on the GEE 21 test include:

1. This graph was made to compare the costs of renting copy machines
from Ames Business Products and from Beck’s Office Supply. What
information is given by the point of intersection of the two lines?
700

600
Total Copies per Month

500

400 Beck
300 Ames

200

100

0
0 2000 4000 6000 8000 10000 12000
Number of Copies Made

a. The number of copies for which the fixed per-month charge is

equal to the cost of copies.
b. The price per copy for renting a copies from both companies.
c. The fixed per-month charge for renting a copier from both
companies.
d. The number of copies for which the total cost is the same for both
companies.

Louisiana GEE 21 Sample Questions, January, 2001

Focused Learning Lessons for Mathematics 40 Algebra

2) Which equation could describe the line of best fit for the graph below?
U. S. Population

300

290

POPULATION (in millions)

280

270

260

250

240

230

220
0 1 2 3 4 5 6 7
YEAR

a) y = 5x + 236
b) y = -5x + 236
1
c) y = x + 236
5
Massachusetts Grade 10 MCAS Re-Test Study Questions, 1999

Attributes of Student Work at the “Got-It” Level

A. When students are graphing equations, they should be able to determine

placement of x- and y-axis, determine the scale for each axis, and
accurately graph two or more points from the equation.

B. Students should be able to interpret a problem situation and accurately

represent it with a linear equation in two variables. Students should be
able to identify which variable represents the x quantity and which
variable represents the y quantity by determining what variable depends
upon the other.

C. Students should be able to use their own, as well as given, linear graphs
to answer questions about problem situations. Students should be able
to interpret the meaning of the x- and y-intercepts, as well as the slope as
it applies to the problem situation.

D. Students should be able to compare two linear graphs and determine the
meaning of the point of intersection.

Focused Learning Lessons for Mathematics 41 Algebra

Lesson 4: Solving and Graphing Linear Equations Teacher Blackline #1

A. x + y = 10

B. In order to play golf at the Scottsville City Golf Course, a person must first join
the club for a one-time membership fee of $125 and then pay a $12 green fee
each time he plays a round of golf.

Focused Learning Lessons for Mathematics 42 Algebra

Lesson 4: Solving and Graphing Linear Equations Teacher Blackline #2

Loan for Used Car

Remaining Debt in

3000
Dollars

2000

1000

0
0 5 10 15
Weeks Since Loan

Focused Learning Lessons for Mathematics 43 Algebra

Lesson 4: Solving and Graphing Linear Equations Student Worksheet

1) Your parents want to throw you a party for your graduation. They have
decided to hire a band for $500 and figure about $15 per person for food
and drink. Write an equation to describe the situation. Then graph the
equation on the grid provided. Be sure to label all parts of the graph. Use
the graph and the equation to determine the cost of inviting 50 people.

2) Use the graph below to answer the following questions

Sally's Pie Shop Daily Profit

40
30
Profit in Dollars

20
10
0
-10 0 5 10 15
-20
-30
Number of Pies Sold

a) How many pies must Sally sell in one day to break even?
b) If Sally has made a profit of $15, how many pies has she sold?
c) How much profit does Sally make on each pie?
d) What does the y-intercept of –30 mean?
e) If Sally sells 50 pies in one day, what will be the profit?

Focused Learning Lessons for Mathematics 44 Algebra

Lesson 4: Solving and Graphing Linear Equations Student Worksheet

3) You are in charge of purchasing the signs for the school’s annual garage
sale. One local company Quick Signs charges a $25 set up fee plus $4 per
sign. Another company Signs of the Times charges a $30 set up fee plus $3
a sign. Write an equation for the cost of purchasing signs from Quick Signs
and write an equation for the cost of purchasing signs from Signs of the
Times.

Graph each equation on the grid below. Label the lines and answer the
following questions.

a. Name the coordinates of the point of intersection. What is the

meaning of that point of intersection?
b. When was Signs of the Times cheaper than Quick Signs?
c. When was Quick Signs cheaper than Signs of the Times?
d. Explain what company you would choose and why.

Focused Learning Lessons for Mathematics 45 Algebra

Lesson 4: Solving and Graphing Linear Equations ANSWERS

Teacher Blackline #1

A. x + y = 10

12

10

Y 4

0
-2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13
-2

-4
X

In order to play golf at the Scottsville City Golf Course, a person must first join
the club for a one-time membership fee of $125 and then pay a $12 green fee
each time he plays a round of golf. y = 125 + 12x

300

250

200

total cost 150

100

50

0
0 2 4 6 8 10 12
number of rounds of golf

Focused Learning Lessons for Mathematics 46 Algebra

Teacher Blackline #2

Y = −200x + 3000

Loan for Used Car

Remaining Debt in

3000
Dollars

2000

1000

0
0 5 10 15
Weeks Since Loan

Things to be discussed from the graph:

1) The original loan was for $3000.
2) The loan will take 15 weeks to be totally paid off
3) The scale on the x-axis is one.
4) The scale on the y-axis is 200.
5) The payment is $200 per week.

Focused Learning Lessons for Mathematics 47 Algebra

Student Worksheet

1) y = 500 + 15x, where y is the cost of the party and x is the number of people in
attendance.

1700
1600
1500
1400
1300
1200
1100
Cost in $

1000
900
800
700
600
500
400
300
200
100
0
0 5 10 15 20 25 30 35 40 45 50 55 60
Number of People

The cost for 50 people is $1250.

2) Use the graph below to answer the following questions

a) Sally must sell 6 pies to break even.

b) For a profit of $15 Sally would have sold 9 pies.
c) Sally makes $5 per pie.
d) It costs Sally $30 just to open her shop. She starts out $30 in the hole.
e) If Sally sells 50 pies in on day, her profit would be $220.

Sally's Pie Shop Daily Profit

40
30
Profit in Dollars

20
10
0
-10 0 5 10 15
-20
-30
Number of Pies Sold

Focused Learning Lessons for Mathematics 48 Algebra

 3) Quick Signs y = 25 + 4x ,
where y is the total cost of the signs and x is the number of signs
Signs of the Times y = 30 + 3x,
where y is the total cost of the signs and x is the number of signs

1. The point of intersection is (5,45). This is the point where the two
companies charge the same for the same number of signs.
2. Signs of the Times is cheaper than Quick Signs for more than 5 signs.
3. Quick Signs is cheaper than Signs of the Times for less than 5 signs.
4. Explanations may vary. If you need more than 5 signs, you would choose
Signs of the Times. If you need less than 5 signs, you would choose Quick
Signs.

90
80
70
60
total cost

50 Quick Signs
40 Signs of the Times
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
number of sign

GEE 21 Connection

1. d
2. a

Focused Learning Lessons for Mathematics 49 Algebra