Linear
Equations
materials modified from Vermont Mathematics Initiative
Equations of Lines
The Film Problem
There are two photography stores in town that do custom film developing, Perfect Picture
(abbreviated PP) and Dynamic Developers (abbreviated DD). At PP the cost to develop one
roll of specialty film is $12 but any additional rolls of film cost only $10. At DD the cost of
developing one roll of film is $24 but each additional roll is developed at a cost of only $8.
For what number of rolls of film is the cost of developing the same at PP and DD?
Method 1: Trial and Error (I.e. “guess and check” or “adaptive guessing.”) As a start, let’s
see what happens with 5 rolls. For PP it will be $12 + 4*$10 = $52, and for DD it will be $24
+ 4*$8 = $56. Close, but not the same. Try 4 rolls. PP = $12 + 3*$10 = $42; and DD = $24 +
3*$8 = $48. Note the gap is larger ($6). So this moved in the wrong direction. Let’s try 7 rolls.
PP = $12 + 6*$10 = $72. DD = $24 + 6*$8 = $72. We have an answer, 7 rolls. Note: With
Trial and Error, we don’t have much insight into whether there might be another answer.
Method 2: Table of Values
rolls of
film
0
1
2
3
4
5
6
7
8
9
10
PP
2
12
22
32
42
52
62
72
82
92
102
DD
16
24
32
40
48
56
64
72
80
88
96
<
This table “invents” a value for 0 rolls by subtracting 10 from the cost of the 1st roll at PP and
8 from the cost of the 1st roll at DD. Does this make sense? Note: Look at what is happening
for 8, 9 or 10 rolls. We can be rather confident that the problem has only one answer.
Method 3: Table with Differences
rolls of
film
0
1
2
3
4
5
6
7
8
9
10
PP
2
12
22
32
42
52
62
72
82
92
102
DD
16
24
32
40
48
56
64
72
80
88
96
Diff
14
12
10
8
6
4
2
0
-2
-4
-6
By keeping track of the differences, we “see” that we are approaching the answer at 7 rolls.
Linear Equations - page 1
Method 4: Mental Arithmetic - We know that DD costs $12 more than PP for one roll but
that for each roll after the first, DD will be $2 less than PP. So we reason that if we developed
6 more rolls (i.e. 7 in all), they will cost the same.
Method 5: Using Algebra – Let n = number of rolls that we purchase. Then the cost for PP is
12 + 10(n – 1) and the cost for DD is 24 + 8(n – 1). Our question is, when are the two amounts
the same? I.e., when is 12 + 10(n – 1) = 24 + 8(n – 1). If we can solve an equation that has
one variable, then we can get an answer of n = 7. To look a bit deeper at this situation, we say
that there is a formula for the cost at PP and a formula for the cost at DD. They are:


PP  10(n  1)  12  10n  2 , the cost (in dollars) of having n rolls of film developed. PP
represents the cost (in dollars).
DD  8(n  1)  24  8n  16 , the cost of having n rolls of film developed.
 $ 
o For DD  8n  16 , we think of 8 as the rate  8
 for having each roll of film
 roll 
developed and we can think of 16 as a “delivery charge.”
$ 

o For PP  10n  2 , 10 is the rate 10
 for having rolls of film developed. The
 roll 
2 can be considered a “delivery charge.”
Method 6: Graph
 We understand that the n is a positive integer (maybe 0, if we want) and because of the
formula, the cost will also be a positive integer. The graph will be a discrete set of points
falling on a line.
The Film Problem
120
100
Cost in $
80
PP
60
DD
40
20
0
0
2
4
6
8
10
12
number of rolls of film
Linear Equations - page 2
Fuel Oil Problem: The Eagle Fuel Company (EFC) charges $1.40 per gallon for Number 2
heating oil and advertises free delivery. The Okemo Petroleum Energy Company (OPEC)
charges $1.25 per gallon of the Number 2 heating oil, but also charges a delivery fee of $20.

The Eagle Fuel Company charges the rate of $1.40 per gallon, written
1.40$
, or
1 gallon
$1.40
$
or 1.40
. We found a formula C  1.40x , which allowed someone to
1 gallon
gallon
compute the cost $C of purchasing x gallons of heating oil.

For Okemo Petroleum Energy Company, we found a formula P  1.25x  20 , which
allowed someone to compute the cost $P of purchasing x gallons of heating oil.

When we graphed this information we found that for EFC the graph is a line in which the
$
rate 1.40
indicates the steepness of the line. One other feature of this line to
gallon
mention is that the line passes through the point where x  0 and C  0 , (the origin of the
coordinate system, where the horizontal and vertical axes cross).

When we graphed this information we also found that for OPEC, the graph is another line
$
in which the rate 1.25
indicates the steepness of the line. This line is not as steep
gallon
as the line for EFC, indicating that OPEC is charging less per gallon for their fuel. This
line for OPEC does not pass through the origin of the coordinate system. When x  0 , the
P  20. This represents the fact that there is a delivery charge of $20. (Someone who
made a call for a delivery, and then told the driver about a change of mind should expect a
charge of $20 for delivery, even though no fuel went into the tank.)

In each case the number representing steepness of the line is referred to as the slope of the
$
line. For EFC, the slope of the line is 1.40 with units
. For OPEC, the slope is
gallon
$
1.25 with units
.
gallon
rise change in y
Slope can be thought of

run change in x

Linear Equations - page 3
Gallons
10
40
60
80
100
120
140
160
180
EFC
14
56
84
112
140
168
196
224
252
OPEC
32.5
70
95
120
145
170
195
220
245
Fuel Oil Problem
300
250
Cost in $
200
EFC
OPEC
Linear (EFC)
Linear (OPEC)
150
y = 1.25x + 20
y = 1.4x
100
50
0
0
50
100
150
200
Gallons of Number 2 Oil
Linear Equations - page 4
Stacking Cups Activity
You have been hired by a company that makes all kinds of cups—foam hot cups, plastic
drinking cups, paper cups, and more—of different sizes. The company needs to know the
measurements of cartons that can hold 50 cups.
1. Make a table and record in it the measurement data (number of cups and height of a stack
of cups) for your cups.
2. Represent the relationship between the number of cups and the height of the stack using a
formula and a graph.
3. Predict how tall a stack of 50 cups would be and explain how you made your prediction.
4. Predict how many cups you would need to create a stack that was approximately 5 feet
tall.
5. Recommend the inside dimensions of a carton that would hold a stack of 50 cups.
6. Transfer your graph and equation on to a large post-it pad. Also, include the dimensions
for a carton that you would recommend to your company that would hold a stack of 50
cups.
7. Compare the results for the different cups. Interpret the meaning of the slope and yintercept with regard to the number and size of the cups, or parts of the cups.
Linear Equations - page 5
Example 1: For the line with equation y  4 x  6 you can interpret this at the county fair.
Let the admission fee be $6 and let the rides cost $4 per ride (a rate!). This formula
y  4 x  6 computes the cost $y of going to the fair and taking x rides.
a) Complete the table below.
b) Graph the line.
c) Find the slope of the line.
d) Find the y-intercept. Interpret the meaning of this point.
e) A point with coordinate (2, v) is on the line. Find v.
f) Interpret what the point in part e) represents with respect to the context of this
problem.
x
y  4x  6
0
6
1
2
3


62
70
Linear Equations - page 6
Example 2: Draining a Bathtub. A bathtub that holds 42 gallons drains at the rate of 4
gallons per 20 seconds. Let W denote the number of gallons of water in the tub x minutes
after the drain is opened.
a) Create a table based on this data that contains at least three points.
b) Graph the points.
c) Write a formula that can be used to compute the number of gallons of water in the tub x
minutes after the drain is opened.
d) Find the slope and interpret the meaning.
e) Find the y-intercept and interpret the meaning.
Example 3: Fred is 3 years older than Jane.
a) Create a table with at least 3 data points showing Fred’s age as a result of Jane’s age.
Jane’s age Fred’s age
b) Sketch a graph showing Fred’s age as a result of Jane’s age for the first 40 years of Jane’s
life.
c) What is the slope of this graph?
d) Mark the point on the graph that corresponds to when Jane was born.
e) Mark the point on the graph when Fred is twice as old as Jane is.
f) If Fred was twice as old as Jane 10 years ago, how old is each of them now?
Linear Equations - page 7
The y  mx  b Form of an Equation of a Line
For the perspective of determining a formula that works in general for all linear problems,
let’s compare the equations that result from two word problems we have done.
I. The Fuel Oil Problem
OPEC charges $1.25 per gallon for fuel oil and they also charge a delivery feel of $20.
delivery fee

P  1.25 x  20

rate
1.25dollars per gallon
II. Draining the Bathtub
We found the equation relating the number of gallons of water in the tub x minutes after the
drain is opened to be:
capacity of tub in gallons

W  (12) x  42

rate
12 gallons per minute
SUMMARY OF THE EXAMPLES
All linear equations have the form:
y -intercept of the line

y 
mx  b

slope of the line
Linear Equations - page 8
Example 4: Let l1 denote a straight line given by the equation y  3x  2 . Let l 2 denote the
straight line given by the equation 2 y  x  12 .
a) Find the slope, y-intercept and x-intercept of l1 .
b) Given that the point (2, y ) is on the line l1 , find y.
c) Find the slope, y-intercept and x-intercept of l 2 .
d) On the same set of axes, graph the lines l1 and l 2 .
e) Find the coordinates of the point of intersection of the lines l1 and l 2 (graphically and
algebraically).
Linear Equations - page 9
Example 5: Given the picture below.
a) Write the equation of the lines l1 and l2.
b) Write an equation of the line that passes through the point (2,5) and is parallel to l1.
c) Write an equation of the line that passes through the point (2,5) and is perpendicular to l1.
Linear Equations - page 10
Simultaneous Linear Equations
Example 6: Use both graphs and algebra to find the points of intersection for each pair of
lines.
3x  y  5
a) 
 x  y  20
 2 x  3 y  11
b) 
4 x  6 y  15
 3x  y  5
c) 
9 x  3 y  15
2 y  x  6
d) 
2 y  4 x  11
Example 7: At a certain store, three pencils and two erasers cost $1.26. At the same store,
six pencils and three erasers cost $2.19. How much is each pencil, and how much is each
eraser?
Linear Equations - page 11