# Linear Equations

```Linear Equations
The Mathematics Readiness Project was funded jointly by the Eisenhower State
Grant Program in Mathematics and Science and the California Academic
Partnership Program.
Overview for the Teacher
Linear Equations
by Katrine Czajkowski
Overview for the Teacher
Consider the following question:
Which of the following is a portion of the graph of y = –2x + 4?
(a)
(b)
(d)
(e)
(c)
The incorrect answer most commonly chosen is (e). Among the reasons
students might miss this question are the following:
1.
2.
3.
4.
5.
6.
7.
8.
Students might not realize that y = –2x + 4 describes the relationship
between two variables.
Students might not recognize that the equation is linear.
Students might not recognize a line graphed on the Cartesian plane as the
graphical representation of a linear equation.
Students might reverse the slope and y-intercept terms.
Students might not understand how to identify the y-intercept of a line.
Students might not understand that slope is simply a way to express the
ratio of the vertical change vs. the horizontal change.
Students might not be able to connect the direction of the line drawn to
the sign of its slope.
Students might not distinguish between a slope of 2 and a slope of –2.
The misunderstandings described above can become particularly acute when
students face a more complex question such as
 x+y=2 
If 
 , then y =
 x–y=6 
(a) –8
(b) –4
(c) –2
(d) 2
(e) 8
The incorrect answer most commonly chosen is (b).
This unit is intended to provide teachers with ways to supplement their
curriculum in order to better prepare students to succeed in their study of
algebra.
Page 2
Section 1: Characteristics of Linear Equations
Section 2: Tables of Values for Graphing
Section 3: Using Intercepts for Graphing
Section 4: Using Slope-Intercept Form with Graphs
Section 5: Families of Linear Equations
Section 6: Graphing Linear vs. Non-Linear Equations
Section 7: Creating, Graphing and Using Linear Equations
Section 8: Simple System of Equations
Section 9: What Went Wrong?
Section 10: Exploring with a Graphing Calculator
Page 3
Section 1: Characteristics of Linear Equations
Teacher Page
For the purpose of this lesson, we will not discuss constant linear equations
(such as x = 7 or y = –10), as they offer exceptions to the rules governing most
linear equations.
Explain to the students that linear equations have several basic characteristics:
1)
2)
3)
The expression contains an equals (=) sign.
The expression contains two variables (usually denoted by "x" and "y").
The expression can be manipulated using addition or subtraction so that
it appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are
constant.
Work through the following examples with the students:
Consider the equation y = 2x + 3. Is this equation linear?
Analysis
1) Does the expression contain an equals (=) sign? Yes.
2) Does the expression contain two variables? Yes.
3) Can the expression be manipulated using addition or subtraction so that it
appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are
constant? Yes, it already looks like y = mx + b.
Is the equation linear? Yes.
Make a table of values where x always changes by the same amount. One
possible response is
x
y
–2
–1
–1
1
0
3
1
5
2
7
3
9
Notice that when the values for x change by the same amount, the values for
y also always change by the same amount.
Page 4
Section 1: Characteristics of Linear Equations
Teacher Page
Consider the equation xy = 8. Is this equation linear?
Analysis
1) Does the expression contain an equals (=) sign? Yes.
2) Does the expression contain two variables? Yes.
3) Can the expression be manipulated using addition or subtraction so that it
appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are
constant? No.
Is the equation linear? No.
Make a table of values where x always changes by the same amount. One
possible response is
x
y
1
6
2
3
3
2
4
1.5
Notice that the values for y do not change by the same amount.
The problems in the student pages of Characteristics of Linear Equations
reinforce these ideas. Students may work individually, in pairs or in groups
of no more than four. Be sure to review their work with them.
I. The equation is linear.
II. The equation is linear.
III. The equations y = 3x + 3, 2y + 5x = 10,
4 = x – 2y and 2x – 3y = 12 are linear.
Page 5
Section 1: Characteristics of Linear Equations
Student Page
I.
Given the equation y = 4 – x, complete the table below.
•
Under "yes/no," check whether or not the equation meets each
characteristic in the first column.
•
In the final column, provide evidence to support your choice of either
"yes" or "no."
Be sure to complete the table of values box. Choose values for x that always
change by the same amount.
In the box at the bottom of the table, answer the question and provide a onesentence reason for your answer.
Characteristics of linear equations
1.
Does
the
expression
contain an equals (=) sign?
2.
Does
the
expression
contain two variables?
3.
Can the expression be
manipulated
using
that it appears like either y
= mx + b or cx + dy = r,
where b, c, d, m and r are
constant?
x
y
Is this equation linear?
Once you have completed this table and reviewed it with your teacher,
proceed to the next page and complete the chart based on the expression
given.
Page 6
Section 1: Characteristics of Linear Equations
Student Page
II. Given the equation x – y = 8, complete the table.
•
Under "yes/no," check whether or not the equation meets each
characteristic in the first column.
•
In the final column, provide evidence to support your choice of either
"yes" or "no."
Be sure to complete the table of values box. Choose values for x that always
change by the same amount.
In the box at the bottom of the table, answer the question and provide a onesentence reason for your answer.
Characteristics of linear equations
1.
Does
the
expression
contain an equals (=) sign?
2.
Does
the
expression
contain two variables?
3.
Can the expression be
manipulated
using
that it appears like either y
= mx + b or cx + dy = r,
where b, c, d, m and r are
constant?
x
y
Is this equation linear?
Once you have completed this table and reviewed it with your teacher,
proceed to the next page and complete the chart based on the expression
given.
Page 7
Section 1: Characteristics of Linear Equations
Student Page
III. Now complete the table below. Check either "linear" or "not linear" for
each equation. In the final column, provide evidence to support your
decision.
Expression
Linear?
Not linear?
y = 3x + 3
2y + 5x = 10
3xy = 12
4 = x – 2y
x2 + 2 = y
2x – 3y = 12
Page 8
Section 2: Tables of Values for Graphing
Teacher Page
Provide the students with a brief review of plotting points in the Cartesian
plane. They then should be able to work through the student pages of Tables
of Values for Graphing. Graph paper should be available for the students.
I.
II.
III.
Page 9
Section 2: Tables of Values for Graphing
I.
Student Page
Consider the equation y = –2x + 2.
Step 1: Make an (x,y) table using at least three of your favorite values for x.
Pick at least one positive value for x, at least one negative value for x, and
use zero as a value for x.
x
y
Step 2: Plot the points on the Cartesian plane.
Step 3: In the plot above, draw a line running through all the points.
Page 10
Section 2: Tables of Values for Graphing
Student Page
For each equation below, construct the (x,y) table of values and graph the
equation on the Cartesian plane. Work with a partner and follow the steps for
using a table of values to graph a linear equation. Show your work in the
space provided.
II. y = 3x – 2
x
y
III. 2x + y = 8
x
y
Page 11
Section 3: Using Intercepts for Graphing
Teacher Page
Work through the graphing of
3x + 4y = 12 with the students.
Do this by first identifying the
x-intercept—(4,0)—and the yintercept—(0,3)—and
then
drawing the line that runs
through these two points.
Then
work
through
the
somewhat
more
complex
1
problem of graphing y = 2 x – 4
by using the x-intercepts and
the y-intercepts.
Have the students work through the Using x-Intercepts and y-Intercepts for
Graphing student pages.
Page 12
Section 3: Using Intercepts for Graphing
Teacher Page
I.
II.
III.
IV.
Page 13
Section 3: Using Intercepts for Graphing
Student Page
The x-intercept is the point where the line crosses the x-axis.
The y-intercept is the point where the line crosses the y-axis.
The two intercepts can be used to quickly graph a linear equation.
I.
Follow the procedure below to graph the equation y = 2x + 4.
Step 1: Find the y-intercept: In the equation substitute the value 0 for x
and solve for y.
Step 2: Fill in the missing number: (0,
) is the y-intercept of y = 2x + 4.
Step 3: Find the x-intercept: In y = 2x + 4 substitute the value 0 for y and
solve for x.
Step 4: Fill in the missing number: (
,0) is the x-intercept of y = 2x + 4.
Step 5: Plot the x-intercept
and the y-intercept on the
Cartesian plane.
Step 6: On the graph above, draw a line running through the two
intercepts.
Page 14
Section 3: Using Intercepts for Graphing
Student Page
For each of the equations below,
a)
b)
c)
d)
e)
Identify the x-intercept of the graph by substituting zero for y.
Plot the x-intercept on the Cartesian plane.
Identify the y-intercept of the graph by substituting zero for x.
Plot the y-intercept on the Cartesian plane.
Draw a line running through the two intercepts.
II. y = 3x – 1
x-intercept: (
,
)
y-intercept: (
,
)
Coordinates of third point:
(
,
)
Does this point lie on the line
you drew after finding the xand y-intercepts?
Why is the
important?
last
question
Page 15
Section 3: Using Intercepts for Graphing
Student Page
III. x – 2y = 8
x-intercept: (
,
)
y-intercept: (
,
)
Coordinates of third point:
(
,
)
Does this point lie on the line
you drew after finding the xand y-intercepts?
Why is the
important?
last
question
Page 16
Section 3: Using Intercepts for Graphing
Student Page
IV. x = 4 + y
x-intercept: (
,
)
y-intercept: (
,
)
Coordinates of third point:
(
,
)
Does this point lie on the line
you drew after finding the xand y-intercepts?
Why is the
important?
last
question
Page 17
Section 4: Using Slope-Intercept Form with Graphs
Teacher Page
In this section you should present to the students the ideas of slope and yintercept. Below we outline for you a possible presentation of these concepts.
Demonstrate for the students
that in a table of values for the
equation y = 2x + 5, the y values
increase by 2 as the x values
increase by 1. Show also that for
any two points on this line, the
ratio of the vertical change
divided by the horizontal
change is 2. Define this number
to be the slope of the line.
Use the equation y = –3x + 1 to
graphically illustrate the idea of
negative slope (–3 in this case).
Point out to the students that
the line represented by the
equation y = –3x + 1 crosses the
y-axis at the point (0,1). This is
called the y-intercept.
Page 18
Section 4: Using Slope-Intercept Form with Graphs
Teacher Page
Use the graph of the equation
y = 2x + 5 to reinforce the idea
of y-intercept. In this case, the
y-intercept is at (0,5)
Summarize for the students the idea that the line represented by an equation
of the form y = mx + b has slope "m" and y-intercept (0,b). We call y = mx + b
the slope-intercept form of the equation of the line.
Have the students work through the Using Slope-Intercept Form with Graphs
student pages. Graph paper should be available for the students.
Page 19
Section 4: Using Slope-Intercept Form with Graphs
Teacher Page
I.
y = –x + 5
II. y = 2x – 4
1
III. y = 2 x + 2
IV. y = –2x – 6
2
V. y = 3 x + 2
VI. y = 3x – 5
Page 20
Section 4: Using Slope-Intercept Form with Graphs
Student Page
In problems I through III, rearrange the terms in the given equation so that
each equation appears in slope-intercept form, y = mx + b. Then graph the
equation by using the y-intercept and the slope.
Remember that
•
The "y" must be alone and positive.
•
The "b" is a constant that is not attached to either variable. The "b"
represents the y-intercept of the line.
•
The "m" is the coefficient of x and represents the slope of the line. The
slope is the ratio of the vertical change to the horizontal in the line from
one point to another.
1.
x+y=5
Slope-intercept form:
Page 21
Section 4: Using Slope-Intercept Form with Graphs
II. 4x – 2y = 8
Slope-intercept form:
III. 3x = 6y – 12
Slope-intercept form:
Student Page
Page 22
Section 4: Using Slope-Intercept Form with Graphs
Student Page
In problems IV through VI, find a linear equation for the line shown. Begin
by finding the y-intercept ("b"). Then use the two given points to identify the
slope ("m"). Finally, write the equation in slope-intercept form.
IV.
b=
m=
Equation:
V.
b=
m=
Equation:
VI.
b=
m=
Equation:
Page 23
Section 5: Families of Linear Equations
Teacher Page
Common errors in graphing linear equations often result from
1.
2.
3.
Confusing the slope (m) with the y-intercept (b).
Inadvertently reversing the sign of the slope.
Plotting the y-intercept on the wrong axis.
If students are alerted to these common errors, they will be better prepared to
avoid them. Studying families of related equations may raise students'
awareness of these potential errors.
Show the students how the
graphs of
y = 2x + 4
and
y = –2x + 4
differ. This will help them
recognize the significance of
the "m" term.
Show the students how the
graphs of
y = 2x + 4
and
y = 2x – 4
differ. This will draw their
attention to the significance of
the "b" term.
Page 24
Section 5: Families of Linear Equations
Teacher Page
Show the students how the
graphs of
y = 2x + 4
and
y = 4x + 2
differ. This will alert them to
the problems associated with
confusing the slope of a line
with its y-intercept.
The comparison charts in the Families of Linear Equations student pages will
help reinforce these important concepts. Students should be encouraged to
create their own comparison charts based on original equations.
Page 25
Section 5: Families of Linear Equations
Teacher Page
I.
II.
III.
Page 26
Section 5: Families of Linear Equations
Student Page
For each of the pairs of equations below, complete the chart. Indicate the slope
and the y-intercept of each equation and graph each line.
Answer the question beneath the graphs using at least one complete sentence.
I.
Equation:
y = 3x + 5
y = –3x + 5
Slope:
y-intercept:
Graph:
What is the reason for the difference in the graphs of the two equations?
Page 27
Section 5: Families of Linear Equations
Student Page
II.
Equation:
y = 2x + 3
y = 2x – 3
Slope:
y-intercept:
Graph:
What is the reason for the difference in the graphs of the two equations?
Page 28
Section 5: Families of Linear Equations
Student Page
III.
Equation:
y = –3x + 2
y = 2x – 3
Slope:
y-intercept:
Graph:
What is the reason for the difference in the graphs of the two equations?
Page 29
Section 6: Graphing Linear vs. Non-Linear Equations
Teacher Page
Use the equation x2 – y = 4 to show the students that not all graphs are lines.
Proceed as follows:
First, generate a table of values for x2 – y = 4:
x
y
–3
5
–2
0
–1
–3
0
–4
1
–3
2
0
3
5
Second, point out that although the x values are increasing by the same
amount, namely 1, the y-values are changing by different amounts. For
example, as x increases from 0 to 1 y increases by 1, while when x increases
from 1 to 2 y increases by 3.
Third, plot the points so that the students will see that the graph is not a line:
Ask students if they know of other characteristics that would indicate that the
graph would not turn out to be a line.
Having the students work through the student pages of Graphing Linear vs.
Non-Linear Equations will reinforce the fact that not all graphs are lines.
Page 30
Section 6: Graphing Linear vs. Non-Linear Equations
Teacher Page
I.
The equation –3x + y = 6 is linear.
II. The equation 2xy = 8 is not linear.
III. The equation 4y = 2x is linear.
Page 31
Section 6: Graphing Linear vs. Non-Linear Equations
Student Page
For each equation,
a) Solve for y in the first box.
b) Create a table of values based on the equation.
c) Graph the equation.
d) Indicate whether or not the equation is linear by writing "yes" or "no"
in the first box.
Equation
Table of Values
–3x + y = 6
x
y
2xy = 8
x
y
4y = 2x
x
y
Graph
Linear?
Linear?
Linear?
Page 32
Section 7: Creating, Graphing and Using Linear Equations
Teacher Page
It is important for students to recognize that equations are just ways of
describing the special relationship that exists between two variables. It is
therefore helpful to provide students with real-world applications of these
ideas.
Work through the following application with the students:
Tickets for the school play go on sale Monday. Prior to this a total of 27 tickets
have been purchased by members of the cast for their families. As the day
progresses a total of 6 tickets are sold each hour. Thus, if "h" represents the
number of hours that have gone by and if "t" represents the number of tickets
that have been sold, we can create a table of values:
h
t
0
27
1
33
2
39
3
45
4
51
We also can write an equation that represents this situation: t = 6h + 27
Finally, we graph this
equation. Draw the
graph only for h
between 0 and 6.
Point out that the
graph is a straight
line.
Show how to use the graph to estimate how many tickets will be sold after 10
The problems in the student pages of Creating, Graphing and Using Linear
Equations provide the students with other applications of linear equations.
Page 33
Section 7: Creating, Graphing and Using Linear Equations
Teacher Page
I.
The equation is B = 15t.
Monique will have 15 tests graded by the
time She has \$225 in her bank account.
II. The equation is I = 5s + 200.
Mr. Gonzalez's income will be \$250 if he
sells 10 shares of stock.
Page 34
Section 7: Creating, Graphing and Using Linear Equations
Student Page
For each situation below,
a) Write an equation that describes the relationship between the two
variables mentioned. State whether or not the equation is linear.
b) Draw the graph of each equation in the space provided.
c) Write two statements that are true based on your graph.
d Use the graph to estimate the amount requested in the final statement.
I.
Monique earns \$15 for every biochemistry test she grades as a teacher's
assistant. She starts the semester with nothing in the bank and saves all of
the money he earns grading tests. Use "B" to represent the amount of
money in her bank account and "t" to represent the number of tests she
Equation:
Statements based on the graph:
many tests Monique will have
graded by the time she has \$225
in her bank account.
Page 35
Section 7: Creating, Graphing and Using Linear Equations
Student Page
II. Each week, Mr. Gonzalez earns \$200 per week plus \$5 for every share of
stock he sells. His income is represented by "I" and the amount of stock
he sells is represented by "s."
Equation:
Statements based on the graph:
Gonzalez's income if he sells 10
shares of stock.
Page 36
Section 8: Simple System of Equations
Teacher Page
Work through the following application with the students:
Thomas has been offered a job where he can earn \$5 for every hour he spends
painting houses. If he starts with nothing in his bank account and doesn't
spend any money all week, how much will be in his bank account?
The answer obviously depends upon how many hours Thomas works. If "h"
represents the number of hours he works and "a" represents the amount of
money in his bank account, we know that a = 5h
Suppose now that Thomas has a alternate job offer in which he would be
given an initial payment of \$12 and then would earn \$3 an hour? Which job
would give Thomas the most money after 10 hours? After 10 hours, the first job
would give him 5× 10 = \$50 and the second job would give him only 3 × 10 + 12 = \$42.
After how many hours of work will both jobs pay him the same amount?
Complete the following table for the first equation, a = 5h:
h
0
1
2
3
4
5
6
7
8
9
a
0
5
10
15
20
25
30
35
40
45
Complete the following table for the second equation, a = 3h + 12:
h
0
1
2
3
4
5
6
7
8
9
a
12
15
18
21
24
27
30
33
36
39
Point out that the pair (6,30) appears in both tables and that 6 hours of work
would pay him the same amount from both jobs.
Draw the graphs of
both equations on the
same axes and point
out that (6,30) is the
point of intersection.
Discuss which job
pays more after 10
hours.
Students should now complete the student pages of Simple Systems of
Equations.
Page 37
Section 8: Simple System of Equations
Teacher Page
I.
(5,3)
II. (1,5)
III. (2,4)
IV. (–1,6)
Page 38
Section 8: Simple System of Equations
I.
Student Page
Consider the system of equations
 x – y = 2

 x + y = 8



Complete the following table for
the first equation, x – y = 2:
Complete the following table for
the second equation, x + y = 8:
x
x
1
2
3
4
5
6
7
y
1
2
3
4
5
6
7
y
What is the common pair (x,y) on both tables?
Verify that this pair (x,y) satisfies the first equation, x – y = 2.
Verify that this pair (x,y) satisfies the second equation, x + y = 8.
To the right, draw the
graphs of both equations,
x – y = 2 and x + y = 8.
At what point (x,y) do the
two lines intersect?
Explain in a full sentence
how this point (x,y) is
related to the (x,y) found
above.
Page 39
Section 8: Simple System of Equations
Student Page
In problems II, III and IV, make a table of values for each of the two given
 x+y=6 
II. If 
 , then y =
 y = 3x + 2 
 y=x+2 
III. If 
 , then x =
 y = 3x – 2 
 2x + y = 4 
IV. If 
 , then y =
 y = 3x + 9 
Page 40
Section 9: What Went Wrong?
Teacher Page
Students can avoid many future mistakes if they have the opportunity to
critique the work of others, especially if that work contains frequently
encountered errors. In this section the students are asked to describe the
mistake and write the correct answer.
Page 41
Section 9: What Went Wrong?
I.
Teacher Page
Amy tried to answer the following question:
Which of the following is a portion of the graph of y = –2x + 4?
(a)
(b)
(d)
(e)
(c)
She wrote (e) for her answer. Explain what Amy did wrong. Use complete
sentences.
II. Max wrote (a) for his answer to the same question. Explain what Max did
wrong. Use complete sentences.
Page 42
Section 10: Exploring with a Graphing Calculator
Teacher Page
If you have graphing calculators available, then students can use them to to
find the intersection of two graphs. Lead the class through the following
graphing calculator activities:
1.
Graph y = 2x – 3 on
the
graphing
calculator. By tracing
verify that the yintercept is at (0,–3)
and that the xintercept is at (1.5,0).
2.
If
the
graphing
calculator
has
a
"table" option, use
the table to view a
listing of (x,y) values
for y = 2x – 3. Use
these
values
to
verify that the slope
of the line is 2.
3.
Graph y = –3x + 2
now on the graphing
calculator,
leaving
the
first
graph,
y = 2x – 3, "on."
x
–1
0
1
2
3
4
y
5
–3
–1
1
3
5
Page 43
Section 10: Exploring with a Graphing Calculator
Teacher Page
4.
By
tracing
and
zooming, verify that
the
point
of
intersection is (1,–1).
5.
If the graphing calculator has a "table" option, use the table to verify that
the coordinates (1,–1) appear on both tables.
x
–1
0
1
2
3
4
y = 2x – 3
5
–3
–1
1
3
5
x
–1
0
1
2
3
4
y = –3x + 2
5
2
–1
–4
–7
–10
Students should now complete the student pages of Exploring with a
Graphing Calculator.
Page 44
Section 10: Exploring with a Graphing Calculator
Teacher Page
III. (1,2)
IV. (–4,–4)
Page 45
Section 10: Exploring with a Graphing Calculator
Student Page
Use your graphing calculator to help do the following problems.
I.
Graph y = –x + 3.
What is the x-intercept? Label
What is the y-intercept? Label
Give the coordinates of three
other points that lie on this
line:
(
,
), (
,
), (
,
)
What is the y-coordinate of the
point with x-coordinate –4?
II. Turn off the graph for #I.
Graph –3x + y = –1.
What is the x-intercept? Label
What is the y-intercept? Label
Give the coordinates of three
other points that lie on this
line:
(
,
), (
,
), (
,
)
Page 46
Section 10: Exploring with a Graphing Calculator
Student Page
III. Turn on the graph for #I,
namely y = –x + 3.
On this set of axes, sketch the
graph of y = –x + 3 and the
graph of –3x + y = –1
What are the coordinates of
the point where the two lines
intersect?
(
,
)
Use "table" mode to view (x,y)
tables for each graph. What
pair of coordinates appears in
the table for both graphs?
(
,
)
1
 y = 2 x – 2 
IV. Use a graphing calculator to graph the equations 
 and
5
 6 – y = – 2 x 
then use the graphs and the calculator functions to identify the solution
to the system.
V. Imagine that your friend has called you on the phone to ask for help
using a graphing calculator to find the solution to a system of two
equations with two variables. On separate paper, write out your
phone, so your directions must be very specific and clear.
Page 47
Section 10: What Went Wrong?
Student Page
Page 48
```