1.1 Linear Models

Investigation

Sales on Commission

Delbert is offered a part-time job selling restaurant equipment. He will be paid $1000 per month plus a 6% commission on his sales. The sales manager tells Delbert he can expect to sell about $8000 worth of equipment per month. To help him decide whether to accept the job, Delbert does a few calculations.

Based on the sales manager’s estimate, what monthly income can Delbert expect from this job? What annual salary would that provide?
What would Delbert’s monthly salary be if he sold only $5000 of equipment per month? What would his salary be if he sold $10,000 worth per month? Compute monthly incomes for each sales total shown in the table.
Sales Income
5000
8000
10,000
12,000
15,000
18,000
20,000
25,000
30,000
35,000
Plot your data points on a graph, using sales, $S$ , on the horizontal axis and income, $I$ , on the vertical axis, as shown in the figure. Connect the data points to show Delbert’s monthly income for all possible monthly sales totals.
Add two new data points to the table by reading values from your graph.
Write an algebraic expression for Delbert’s monthly income, $I$ , in terms of his monthly sales, $S$ . Use the description in the problem to help you:
He will be paid: $1000... plus a 6% commission on his sales.
Income $= \underset{―}{}$
Test your formula from part (5) to see if it gives the same results as those you recorded in the table.
Use your formula to find out what monthly sales total Delbert would need in order to have a monthly income of $2500.
Each increase of $1000 in monthly sales increases Delbert’s monthly income by _______________.
Summarize the results of your work: In your own words, describe the relationship between Delbert’s monthly sales and his monthly income. Include in your discussion a description of your graph.

Tables, Graphs and Equations

The first step in creating a model is to describe relationships between variables. In Sales on Commission, we analyzed the relationship between Delbert's sales and his income. Starting from a verbal description, we represented the relationship in three different ways.

A table of values displays specific data points with precise numerical values.
A graph is a visual display of the data. It is easier to spot trends and describe the overall behavior of the variables from a graph.
An algebraic equation is a compact summary of the model. It can be used to analyze the model and to make predictions

We begin our study of modeling with some examples of linear models. In the examples that follow, observe the interplay among the three modeling tools, and how each contributes to the model.

Example 1

Annelise is on vacation at a seaside resort. She can rent a bicycle from her hotel for $3 an hour, plus a $5 insurance fee. (A fraction of an hour is charged as the same fraction of $3.)

Make a table of values showing the cost, $C$ , of renting a bike for various lengths of time, $t$ .
Plot the points on a graph. Draw a curve through the data points.
Write an equation for $C$ in terms of $t$ .

To find the cost, we multiply the time by $3, and add the result to the $5 insurance fee. For example, the cost of a 1-hour bike ride is

$\begin{aligned} Cost & = ($ 5 insurance fee) + ($ 3 per hour) \times (1 hour) \\ C & = 5 + 3 (1) = 8 \end{aligned}$

A 1-hour bike ride costs $8. We record the results in a table, as shown here:

Length of rental (hours)	Cost of rental (dollars)		$(t, C)$
$1$	$8$	$C = 5 + 3 (1)$	$open parenthesis 1, 8 close parenthesis$
$2$	$11$	$C = 5 + 3 (2)$	$open parenthesis 2, 11 close parenthesis$
$3$	$14$	$C = 5 + 3 (3)$	$open parenthesis 3, 14 close parenthesis$

Each pair of values represents a point on the graph. The first value gives the horizontal coordinate of the point, and the second value gives the vertical coordinate. The points lie on a straight line, as shown in the figure. The line extends infinitely in only one direction, because negative values of $t$ do not make sense here.
To write an equation, we let $C$ represent the cost of the rental, and we use $t$ for the number of hours:
$\begin{aligned} Cost & = ($ 5 insurance fee) + ($ 3 per hour) \times (number of hours) \\ C & = 5 + 3 \cdot t \end{aligned}$

When you graph the data given in a table, on which axis do you show the variable in the first row of the table?

_____

The horizontal axis

When you graph the data given in a table, on which axis do you show the variable in the first row of the table?

The linear axis
The horizontal axis
The vertical axis
Both axes

Example 2

Use the equation $C = 5 + 3 \cdot t$ you found in Example to answer the following questions. Then show how to find the answers by using the graph.

How much will it cost Annelise to rent a bicycle for 6 hours?
How long can Annelise bicycle for $18.50?

We substitute $t equals 6$ into the expression for $C$ to find
$C = 5 + 3 (6) = 23$
A 6-hour bike ride will cost $23. The point $P$ on the graph in the figure represents the cost of a 6-hour bike ride. The value on the $C$ -axis at the same height as point $P$ is 23, so a 6-hour bike ride costs $23.
We substitute $C equals 18.50$ into the equation and solve for $t$ .
$\begin{aligned} 18.50 & = 5 + 3 t \\ 13.50 & = 3 t \\ t & = 4.5 \end{aligned}$
For $18.50 Annelise can bicycle for 4½ hours. The point $Q$ on the graph represents an $18.50 bike ride. The value on the $t$ -axis below point $Q$ is 4.5, so $18.50 will buy a 4.5 hour bike ride.

Frank plants a dozen corn seedlings, each 6 inches tall. With plenty of water and sunlight they will grow approximately 2 inches per day. Complete the table of values for the height, $h$ , of the seedlings after $t$ days.

Complete the table of values for the height, $h$ of the seedlings after $t$ days.

$t$	$0$	$5$	$10$	$15$	$20$
$h$	_____	_____	_____	_____	_____

Write an equation for the height $h$ of the seedlings in terms of the number $t$ of days since they were planted.
Equation: _____
Graph the equation.

$t$	$0$	$5$	$10$	$15$	$20$
$h$	h0	h1	h2	h3	h4

$h = 6 + 2 t$
The graph of seedling height vs time is shown below.

The graph of seedling height vs time for part (b) is shown below.

Complete the table of values for the height, $h$ of the seedlings after $t$ days.

$t$	$0$	$5$	$10$	$15$	$20$
$h$

Write an equation for the height $h$ of the seedlings in terms of the number $t$ of days since they were planted.
Graph the equation.

$h = 6 + 2 t$

Use your equation from Practice 1 to answer the questions. Illustrate each answer on the graph.

How tall is the corn after Tweek weeks? Use "ft" for feet or "in" for inches.
Answer (including units): _____
How long will it be before the corn is Hft feet tall? Use "day" for days.
Answer (including units): _____

48 inches tall
33 days

A graph is below.

The graph below illustrates the answers.

Use your equation from Practice 1 to answer the questions. Illustrate each answer on the graph.

How tall is the corn after 3 weeks?
How long will it be before the corn is 6 feet tall? Hint: Convert feet to inches.

48 inches tall
33 days

What is the difference between an expression and an equation?

_____

What is the difference between an expression and an equation?

Choosing Scales for the Axes

To create a useful graph, we must choose appropriate scales for the axes.

The axes must extend far enough to show the values of the variables.
The tick marks should be equally spaced.
Usually we should use no more than 10 or 15 tick marks.

Example 3

In 1990, the median price of a home in the US was $92,000. The median price increased by about $4700 per year over the next decade.

Make a table of values showing the median price of a house in 1990, 1994, 1998, and 2000.
Choose suitable scales for the axes and plot the values you found in part (a) on a graph. Use $t$ , the number of years since 1990, on the horizontal axis and the price of the house, $P$ , on the vertical axis. Draw a curve through the points.
Write an equation that expresses $P$ in terms of $t$ .
How much did the price of the house increase from 1990 to 1996? Illustrate the increase on your graph.

In 1990 the median price was $92,000. Four years later, in 1994, the price had increased by

4 open parenthesis 4700 close parenthesis equals 18, 800

dollars, so

$P = 92, 000 + 4 (4700) = 110, 800$

In 1998 the price had increased by

8 open parenthesis 4700 close parenthesis equals 37, 600

dollars, so

$P = 92, 000 + 8 (4700) = 129, 600$

You can verify the price of the house in 2000 by a similar calculation.

Year	Price of House)	$(t, P)$
$1990$	$92, 000$	$open parenthesis 0, 92, 000 close parenthesis$
$1994$	$110, 800$	$open parenthesis 4, 110, 800 close parenthesis$
$1998$	$129, 600$	$open parenthesis 8, 129, 600 close parenthesis$
$2000$	$139, 000$	$open parenthesis 10, 139, 000 close parenthesis$

We let $t$ stand for the number of years since 1990, so that $t equals 0$ in 1990, $t equals 4$ in 1994, and so on. To choose scales for the axes, we look at the values in the table. For this graph we scale the horizontal axis, or $t$ -axis, in 1-year intervals and the vertical axis, or $P$ -axis, for $90,000 to $140,000 in intervals of $5,000. The points lie on a straight line, as shown in the figure.
Look back at the calculations in part (a). The price of the house started at $92,000 in 1990 and increased by $t times 4700$ dollars after $t$ years. Thus,
$P = 92, 000 + 4700 t$
We find the points on the graph for 1990 and 1996. These points lie above $t equals 0$ and $t equals 6$ on the $t$ -axis. Next we find the values on the $P$ -axis corresponding to the two points. The values are $P equals 92, 000$ in 1990 and $P equals 120, 200$ in 1996. The increase in price is the difference of the two $P$ -values.
$\begin{aligned} increase in price & = 120, 200 - 92, 000 \\ = 28, 200 \end{aligned}$
The price of the home increased $28,200 between 1990 and 1996. This increase is indicated by the arrows in the figure.

If $C$ is expressed in terms of $H$ , which variable goes on the horizontal axis?

_____

$H$

If $C$ is expressed in terms of $H$ , which variable goes on the horizontal axis?

Silver Lake has been polluted by industrial waste products. The concentration of toxic chemicals in the water is currently 285 parts per million (ppm). Environmental officials would like to reduce the concentration by 15 ppm each year.

Complete the table of values showing the desired concentration,

C

, of toxic chemicals

t

years from now. For each

t

-value, calculate the corresponding value for

C

. Write your answers as ordered pairs.

$t$		$(t, C)$
$0$	$C = 285 - 15 (0)$	(0, _____ )
$5$	$C = 285 - 15 (5)$	(5, _____ )
$10$	$C = 285 - 15 (10)$	(10, _____ )
$15$	$C = 285 - 15 (15)$	(15, _____ )

To choose scales for the axes, notice that the value of $C$ starts at 285 and decreases from there. We'll scale the vertical axis up to 300, and use 10 tick marks at intervals of 30. Graph the ordered pairs on the grid, and connect them with a straight line.
Write an equation for the concentration, $C$ , of toxic chemicals $t$ years from now.
Equation: _____

For part (c): The concentration is initially 285 ppm, and we subtract 15 ppm for each year that passes, or $15 times t$ .

$(t, C)$
$open parenthesis 0, 285 close parenthesis$
$open parenthesis 5, 210 close parenthesis$
$open parenthesis 10, 135 close parenthesis$
$open parenthesis 15, 60 close parenthesis$
The graph is shown below.
$C = 285 - 15 t$

The graph for part(b):

Complete the table of values showing the desired concentration,

C

, of toxic chemicals

t

years from now. For each

t

-value, calculate the corresponding value for

C

. Write your answers as ordered pairs.

$t$	$C$	$(t, C)$
$0$	$285 minus 15 open parenthesis 0 close parenthesis$	$open parenthesis 0, close parenthesis$
$5$	$285 minus 15 open parenthesis 5 close parenthesis$	$open parenthesis 5, close parenthesis$
$10$	$285 minus 15 open parenthesis 10 close parenthesis$	$open parenthesis 10, close parenthesis$
$15$	$285 minus 15 open parenthesis 15 close parenthesis$	$open parenthesis 15, close parenthesis$

To choose scales for the axes, notice that the value of $C$ starts at 285 and decreases from there. We'll scale the vertical axis up to 300, and use 10 tick marks at intervals of 30. Graph the ordered pairs on the grid, and connect them with a straight line.
Write an equation for the concentration, $C$ , of toxic chemicals $t$ years from now. Hint: The concentration is initially 285 ppm, and we subtract 15 ppm for each year that passes, or $15 times t$ .

$(t, C)$
$open parenthesis 0, 285 close parenthesis$
$open parenthesis 5, 210 close parenthesis$
$open parenthesis 10, 135 close parenthesis$
$open parenthesis 15, 60 close parenthesis$
$C = 285 - 15 t$

If $x greater than 5$ , what is true about $minus 2 x$ ?

_____

$minus 2 x$ is less than $minus 10$ .

If $x greater than 5$ , what is true about $minus 2 x$ ?

It is greater than 3
It is less than 3
It is greater than 10
It is less than $minus 10$

Example 4 Using a Graphing Utility

In Example, we found the equation

$P = 92, 000 + 4700 t$

for the median price of a house $t$ years after 1990. Graph this equation with technology.

To begin, we press Y= and enter

$Y 1 = 92, 000 + 4700 X$

For this graph, we’ll use the grid in Example for our window settings, so we press WINDOW and enter

Xmin $equals 0$		Xmax $equals 10$
Ymin $equals 90, 000$		Ymax $equals 140, 000$

Finally, we press GRAPH. The graph is shown in the figure.

Solve the equation $2 y - 1575 = 45 x$ for $y$ in terms of $x$ .
$y =$ _____
Graph the equation with a graphing utility. Use the window
Xmin $= - 50$ Xmax $equals 50$ Xscl $equals 5$
Ymin $= - 500$ Ymax $equals 1000$ Yscl $equals 100$
Sketch the graph on paper. Use the window settings to choose appropriate scales for the axes.

Xmin $= - 50$		Xmax $equals 50$		Xscl $equals 5$
Ymin $= - 500$		Ymax $equals 1000$		Yscl $equals 100$

$y = (1575 + 45 x) / 2$
The calculator graph is shown below.
The graph is shown below.

The graph for (b):

Solve the equation $2 y - 1575 = 45 x$ for $y$ in terms of $x$ .
Graph the equation with a graphing utility. Use the window
Xmin $= - 50$ Xmax $equals 50$ Xscl $equals 5$
Ymin $= - 500$ Ymax $equals 1000$ Yscl $equals 100$
Sketch the graph on paper. Use the window settings to choose appropriate scales for the axes.

Xmin $= - 50$		Xmax $equals 50$		Xscl $equals 5$
Ymin $= - 500$		Ymax $equals 1000$		Yscl $equals 100$

$y = (1575 + 45 x) / 2$

What is wrong with the grid above?

_____

What is wrong with the grid above?

The grid lines on the $x$ -axis are not evenly spaced.
The scale on the $y$ -axis does not start at 0.
The axes are not labeled with the variables.
All of the above.

Linear Equations

All the models in the preceding examples have equations with a similar form:

$y = (starting value) + (rate of change) \cdot x$

(We'll talk more about rate of change in Slope and Rate of Change.) Their graphs were all portions of straight lines. For this reason such equations are called linear equations. The order of the terms in the equation does not matter. For example, the equation in Example,

$C = 5 + 3 t$

can be written equivalently as

$- 3 t + C = 5$

and the equation in Example,

$P = 92, 000 + 4700 t$

can be written as

$- 4700 t + P = 92, 000$

This form of a linear equation, $A x + B y = C$ , is called the general form.

Example 5

The manager at Albert's Appliances has $3000 to spend on advertising for the next fiscal quarter. A 30-second spot on television costs $150 per broadcast, and a 30-second radio ad costs $50.

The manager decides to buy $x$ television ads and $y$ radio ads. Write an equation relating $x$ and $y$ .
Make a table of values showing several choices for $x$ and $y$ .
Plot the points from your table, and graph the equation.

Each television ad costs $150, so $x$ ads will cost $ $150 x$ . Similarly, $y$ radio ads will cost $ $50 y$ . The manager has $3000 to spend, so the sum of the costs must be $3000. Thus,
$150 x + 50 y = 3000$
We choose some values of $x$ , and solve the equation for the corresponding value of $y$ . For example, if $x equals 10$ then
$\begin{aligned} 150 (10) + 50 y & = 300 \\ 1500 + 50 y & = 3000 \\ 50 y & = 1500 \\ y & = 30 \end{aligned}$
If the manager buys 10 television ads, she can also buy 30 radio ads. You can verify the other entries in the table.
$x$ $8$ $10$ $12$ $14$
$y$ $36$ $30$ $24$ $18$
We plot the points from the table. All the solutions lie on a straight line, as shown in the figure.

In central Nebraska, each acre of corn requires 25 acre-inches of water per year, and each acre of winter wheat requires 18 acre-inches of water. (An acre-inch is the amount of water needed to cover one acre of land to a depth of one inch.) A farmer can count on 9000 acre-inches of water for the coming year. (Source: Institute of Agriculture and Natural Resources, University of Nebraska)

Write an equation relating the number of acres of corn, $x$ , and the number of acres of wheat, $y$ , that the farmer can plant.
_____
Complete the table. Round your answers to tenths.
$x$ $50$ $100$ $150$ $200$
$y$ _____ _____ _____ _____

$x$	$50$	$100$	$150$	$200$
$y$	_____	_____	_____	_____

$25 x + 18 y = 9000$
$x$ $50$ $100$ $150$ $200$
$y$ $430.6$ $361.1$ $291.7$ $222.2$

$x$	$50$	$100$	$150$	$200$
$y$	$430.6$	$361.1$	$291.7$	$222.2$

Write an equation relating the number of acres of corn, $x$ , and the number of acres of wheat, $y$ , that the farmer can plant.

Complete the table. Round your answers to tenths.

$x$	$50$	$100$	$150$	$200$
$y$

$25 x + 18 y = 9000$
$x$ $50$ $100$ $150$ $200$
$y$ $430.6$ $361.1$ $291.7$ $222.2$

$x$	$50$	$100$	$150$	$200$
$y$	$430.6$	$361.1$	$291.7$	$222.2$

Write down two different equation forms for linear models. Which do you think is easier to use?

_____

Write down two different equation forms for linear models. Which do you think is easier to use?

Intercepts

Consider the graph of the equation

$3 x - 4 y = 12$

shown at left. The points where the graph crosses the axes are called the intercepts of the graph. The coordinates of these points are easy to find.

The $y$ -coordinate of the $x$ -intercept is zero, so we set $y equals 0$ in the equation to get

$\begin{aligned} 3 (0) - 4 x & = 12 \\ x = - 3 \end{aligned}$

The $x$ -intercept is the point $open parenthesis minus 3, 0 close parenthesis$ . Also, the $x$ -coordinate of the $y$ -intercept is zero, so we set $x equals 0$ in the equation to get

$\begin{matrix} 3 y - 4 (0) = 12 \\ y = 4 \end{matrix}$

The $y$ -intercept is $open parenthesis 0, 4 close parenthesis$ .

What is the $y$ -coordinate of any point on the $x$ -axis?

_____

$0$

What is the $y$ -coordinate of any point on the $x$ -axis?

$(x, y)$
$y$
It depends on the value of $x$ .
$0$

The intercepts of a graph tell us something about the situation it models.

Example 6

Find the intercepts of the graph in Practice 3, about the pollution in Silver Lake.
Find the intercepts of the graph in Practice 3, about the pollution in Silver Lake.
What do the intercepts tell us about the problem?

An equation for the concentration of toxic chemicals is
$C = 285 - 15 t$
To find the $C$ -intercept, we set $t$ equal to zero.
$C = 285 - 15 (0) = 285$
The $C$ -intercept is the point $open parenthesis 0, 285 close parenthesis$ , or simply $285$ .
To find the $t$ -intercept, we set $C$ equal to zero and solve for $t$ .
$\begin{aligned} 0 & = 285 - 15 t & Add 15 t to both sides. \\ 15 t & = 285 & Divide both sides by 15. \\ t & = 19 \end{aligned}$

The $t$ -intercept is the point $open parenthesis 19, 0 close parenthesis$ , or simply $19$ .
The $C$ -intercept represents the concentration of toxic chemicals in Silver Lake now: When $t equals 0$ , $C equals 285$ , so the concentration is currently $285$ ppm.
The $t$ -intercept represents the number of years it will take for the concentration of toxic chemicals to drop to zero: When $C equals 0$ , $t equals 19$ , so it will take $19$ years for the pollution to be eliminated entirely.

Delbert says that the intercepts of the line $3 x + 5 y = 30$ are $open parenthesis 10, 6 close parenthesis$ . What is wrong with his answer?

_____

All of the above

Delbert says that the intercepts of the line $3 x + 5 y = 30$ are $open parenthesis 10, 6 close parenthesis$ . What is wrong with his answer?

$open parenthesis 10, 6 close parenthesis$ is not on the $x$ -axis.
An intercept must have a 0 coordinate.
The line has two intercepts.
All of the above.

Find the intercepts of the graph in Example, about the advertising budget for Albert's Appliances: $150 x + 50 y = 3000$ .

Enter each intercept as an ordered pair.
The $x$ -intercept is _____.
The $y$ -intercept is _____.
What do the intercepts tell us about the problem?
The $x$ -intercept tells us:
_____
The $y$ -intercept tells us:
_____

We find the $x$ -intercept by setting $y equals 0$ and solving for $x$ to learn that $x equals x0$ , so the $x$ -intercept is $xint$ .
We find the $y$ -intercept by setting $x equals 0$ and solving for $y$ to learn that $y equals y0$ , so the $y$ -intercept is $yint$ .
The $x$ -intercept has $y equals 0$ , that is, it corresponds to when there are zero radio ads. The $y$ -intercept has $x equals 0$ , that is, it corresponds to when there are zero tv ads.

Find the intercepts of the graph in Example, about the advertising budget for Albert's Appliances: $150 x + 50 y = 3000$ .
What do the intercepts tell us about the problem?

The $x$ -intercept is $open parenthesis 20, 0 close parenthesis$ . The $y$ -intercept is $open parenthesis 0, 60 close parenthesis$ .
The manager can buy 20 television ads if she buys no radio ads. She can buy 60 radio ads if she buys no television ads.

Explain how the words intercept and intersect are related, and how they are different.

_____

Explain how the words intercept and intersect are related, and how they are different.

Intercept Method for Graphing Lines

Because we really only need two points to graph a linear equation, we might as well find the intercepts first and use them to draw the graph. The values of the intercepts will also help us choose suitable scales for the axes. It is always a good idea to find a third point as a check.

Example 7

Find the $x$ - and $y$ -intercepts of the graph of $150 x - 180 y = 9000$ .
Use the intercepts to graph the equation. Find a third point as a check.

To find the $x$ -intercept, we set $y equals 0$ .
$\begin{aligned} 150 x - 18 (0) & = 9000 & Simpify. \\ 150 x & = 9000 & Divide both sides by 150. \\ x & = 60 \end{aligned}$

The $x$ -intercept is the point $open parenthesis 60, 0 close parenthesis$ . To find the $y$ -intercept, we set $x equals 0$ .
$\begin{aligned} 150 (0) - 18 y & = 9000 & Simpify. \\ - 180 y & = 9000 & Divide both sides by - 180 . \\ y & = - 50 \end{aligned}$

The $y$ -intercept is the point $open parenthesis 0, minus 50 close parenthesis$ .
We scale both axes in intervals of 10 and then plot the two intercepts, $open parenthesis 60, 0 close parenthesis$ and $open parenthesis 0, minus 50 close parenthesis$ . We draw the line through them, as shown below. Finally, we find another point and check that it lies on this line. We choose $x equals 20$ and solve for $y$ .
$\begin{aligned} 150 (20) - 180 y & = 9000 \\ 3000 - 180 y & = 9000 \\ - 180 y & = 6000 \\ y & = - 33. \overset{―}{3} \end{aligned}$
We plot the point $(20, - 33 \frac{1}{3})$ . Because this point lies on the line, we can be reasonably confident that our graph is correct.

Is it possible for the $x$ -intercept and the $y$ -intercept of a line to be the same point?

_____

Yes

Is it possible for the $x$ -intercept and the $y$ -intercept of a line to be the same point?

How many points do you need to graph a linear equation?

_____

Two

How many points do you need to graph a linear equation?

Two
Three
One in each quadrant
It depends on the equation

Technology

Choosing a Graphing Window

Knowing the intercepts can also help us choose a suitable window on a graphing utility. We would like the window to be large enough to show the intercepts. For the graph in the example above, we can enter the equation

$Y = (9000 - 150 X) / - 180$

in the window

Xmin $= - 20$		Xmax $equals 70$
Ymin $= - 70$		Ymax $equals 30$

In Practice 5 you wrote an equation about crops in Nebraska.

Find the intercepts of the graph.
Note: Enter each intercept as an ordered pair.
The $x$ -intercept is _____.
The $y$ -intercept is _____.
Use the intercepts to help you choose appropriate scales for the axes, and then graph the equation.
What do the intercepts tell us about the problem?
The $x$ -intercept tells us:
_____
The $y$ -intercept tells us:
_____

We find the $x$ -intercept by setting $y equals 0$ and solving for $x$ to learn that $x equals x0$ , so the $x$ -intercept is $xint$ .
We find the $y$ -intercept by setting $x equals 0$ and solving for $y$ to learn that $y equals y0$ , so the $y$ -intercept is $yint$ .
If he plants no wheat, the farmer can plant 360 acres of corn. If he plants no corn, the farmer can plant 500 acres of wheat.

A graph is shown below.

In Practice 5 you wrote an equation about crops in Nebraska.

Find the intercepts of the graph.
Use the intercepts to help you choose appropriate scales for the axes, and then graph the equation.
What do the intercepts tell us about the crops?

The $x$ -intercept is $open parenthesis 360, 0 close parenthesis$ . The $y$ -intercept is $open parenthesis 0, 500 close parenthesis$ .
If he plants no wheat, the farmer can plant 360 acres of corn. If he plants no corn, the farmer can plant 500 acres of wheat.

What was the most difficult part of this section to understand? Write a question whose answer would help you understand it.

_____

What was the most difficult part of this section to understand? Write a question whose answer would help you understand it.

Section Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

Variable
Solve an equation
Evaluate an expression
Linear equation
Increasing graph
Decreasing graph
Intercept
Mathematical model

CONCEPTS

We can describe a relationship between variables with a table of values, a graph, or an equation.
Linear models have equations of the following form:
$y = (starting value) + (rate of change) \cdot x$
To make a useful graph, we must choose appropriate scales for the axes.
Note
General Form for a Linear Equation
The graph of any equation
$A x + B y = C$
where $A$ and $B$ are not both equal to zero, is a straight line.
The intercepts of a graph are the points where the graph crosses the axes.
We can use the intercepts to graph a line.
Note
To Graph a Line Using the Intercept Method:
1. Find the intercepts of the line.
  1. To find the $x$ -intercept, set $y equals 0$ and solve for $x$ .
  2. To find the $y$ -intercept, set $x equals 0$ and solve for $y$ .
2. Plot the intercepts.
3. Choose a value for $x$ and find a third point on the line.
4. Draw a line through the points.
The intercepts are also useful for interpreting a model.

STUDY QUESTIONS

Name three ways to represent a relationship between two variables.
If $C$ is expressed in terms of $H$ , which variable goes on the horizontal axis?
Explain the difference between evaluating an expression and solving an equation.
How many points do you need to graph a linear equation?
Explain how the words intercept and intersect are related; explain how they are different.
Delbert says that the intercepts of the line $3 x + 5 y = 30$ are $open parenthesis 10, 6 close parenthesis$ . What is wrong with his answer?

SKILLS

Practice each skill in the Homework problems listed.

Make a table of values: #1–4, 7 and 8
Plot points and draw a graph: #1–4, 7 and 8
Choose appropriate scales for the axes: #5–12
Write a linear model of the form $y = (starting value) + (rate of change) \cdot x$ : #1–8
Write a linear model in general form: #25–28, 33–36
Evaluate a linear expression, algebraically and graphically: #1–4
Solve a linear equation, algebraically and graphically: #1–4
Find the intercepts of a graph: #5 and 6, 13–24, 45–52
Graph a line by the intercept method: #5 and 6, 13–24
Interpret the meaning of the intercepts: #5 and 6, 25–28
Use a graphing calculator to graph a line: #37–52
Sketch on paper a graph obtained on a calculator: #37–44

Homework 1.1

The temperature in the desert at 6 a.m., just before sunrise, was $65 degrees$ F. The temperature rose $5$ degrees every hour until it reached its maximum value at about 5 p.m. Complete the table of values for the temperature, $T$ , at $h$ hours after 6 a.m.

$h$	$0$	$3$	$6$	$9$	$10$
$T$

Write an equation for the temperature, $T$ , in terms of $h$ .
Graph the equation.
How hot is it at noon? Illustrate the answer on your graph.
When will the temperature be $110 degrees$ F? Illustrate the answer on your graph.

$h$	$0$	$3$	$6$	$9$	$10$
$T$	$65$	$80$	$95$	$110$	$115$

$T = 65 + 5 h$
$95 degrees$
3 p.m.

The taxi out of Dulles Airport charges a traveler with one suitcase an initial fee of $ $2.00$ , plus $ $1.50$ for each mile traveled. Complete the table of values showing the charge, $C$ , for a trip of $n$ miles.

$n$	$0$	$5$	$10$	$15$	$20$	$25$
$C$

Write an equation for the charge, $C$ , in terms of the number of miles traveled, $n$ .
Graph the equation.
What is the charge for a trip to Mount Vernon, $40$ miles from the airport? Illustrate the answer on your graph.
If a ride to the National Institutes of Health (NIH) costs $ $39.50$ , how far is it from the airport to the NIH? Illustrate the answer on your graph.

On October 31, Betty and Paul fill their $250$ -gallon oil tank for their heater. Beginning in November, they use an average of $15$ gallons of oil per week. Complete the table of values for the amount of oil, $A$ , left in the tank after $w$ weeks.

$w$	$0$	$4$	$8$	$12$	$16$
$A$

Write an equation that expresses the amount of oil, $A$ , in the tank in terms of the number of weeks, $w$ , since October 31.
Graph the equation.
How much did the amount of fuel oil in the tank decrease between the third week and the eighth week? Illustrate this amount on the graph.
When will the tank contain more than $175$ gallons of fuel oil? Illustrate on the graph.

$w$	$0$	$4$	$8$	$12$	$16$
$A$	$250$	$190$	$130$	$70$	$10$

$A = 250 - 15 w$
75 gallons
Until the fifth week

Leon's camper has a $20$ -gallon gas tank, and he gets $12$ miles to the gallon. (That is, he uses $\frac{1}{12}$ gallon per mile.) Complete the table of values for the amount of gas, $g$ , left in Leon's tank after driving $m$ miles.

$m$	$0$	$48$	$96$	$144$	$192$
$g$

Write an equation that expresses the amount of gas, $g$ , in Leon's fuel tank in terms of the number of miles, $m$ , he has driven.
Graph the equation.
How much gas will Leon use between 8 a.m., when his odometer reads $96$ miles, and 9 a.m., when the odometer reads $144$ miles? Illustrate on the graph.
If Leon has less than $5$ gallons of gas left, how many miles has he driven? Illustrate on the graph.

Phil and Ernie buy a used photocopier for $ $800$ and set up a copy service on their campus. For each hour that the copier runs, Phil and Ernie make $ $40$ .

Write an equation that expresses Phil and Ernie's profit (or loss), $P$ , in terms of the number of hours, $t$ , they run the copier.
Find the intercepts and sketch the graph. (Suggestion: Scale the horizontal axis from $0$ to $40$ in increments of $5$ , and scale the vertical axis from $minus 1000$ to $400$ in increments of $100$ .)
What do the intercepts tell us about the profit?

$P = - 800 + 40 t$
$open parenthesis 0, minus 800 close parenthesis$ , $open parenthesis 20, 0 close parenthesis$
The $P$ -intercept, $minus 800$ , is the initial $open parenthesis t equals 0 close parenthesis$ value of the profit. Phil and Ernie start out $$ 800$ in debt. The $t$ -intercept, $20$ , is the number of hours required for Phil and Ernie to break even.

A deep-sea diver is taking some readings at a depth of $400$ feet. He begins rising at $20$ feet per minute.

Write an equation that expresses the diver’s altitude, $h$ , in terms of the number of minutes, $m$ , elapsed. (Consider a depth of $400$ feet as an altitude of $minus 400$ feet.)
Find the intercepts and sketch the graph. (Suggestion: Scale the horizontal axis from $0$ to $24$ in increments of $2$ , and scale the vertical axis from $minus 500$ to $100$ in increments of $50$ .)
What do the intercepts tell us about the diver's depth?

There are many formulas for estimating the annual cost of driving. The Automobile Club estimates that fixed costs for a small car—including insurance, registration, depreciation, and financing—total about $ $5000$ per year. The operating costs for gasoline, oil, maintenance, tires, and so forth are about $12.5$ cents per mile. (Source: Automobile Association of America)

Write an equation for the annual driving cost, $C$ , in terms of $d$ , the number of miles driven.

Complete the table of values.

Miles Driven	$4000$	$8000$	$12, 000$	$16, 000$	$20, 000$
Cost ($)

Choose scales for the axes and graph the equation.
How much does the annual cost of driving increase when the mileage increases from $8000$ to $12, 000$ miles? Illustrate this amount on the graph.
How much mileage will cause the annual cost to exceed $ $7000$ ? Illustrate on the graph.

$C = 5000 + 0.125 d$
Complete the table of values.
Miles Driven $4000$ $8000$ $12, 000$ $16, 000$ $20, 000$
Cost ($) $5500$ $6000$ $6500$ $7000$ $7500$
$ $500$
More than 16,000 miles

Miles Driven	$4000$	$8000$	$12, 000$	$16, 000$	$20, 000$
Cost ($)	$5500$	$6000$	$6500$	$7000$	$7500$

The boiling point of water changes with altitude. At sea level, water boils at $212 degrees$ F, and the boiling point diminishes by approximately $0.002 degrees$ F for each $1$ -foot increase in altitude.

Write an equation for the boiling point, $B$ , in terms of $a$ , the altitude in feet.

Complete the table of values.

Altitude (ft)	$minus 500$	$0$	$1000$	$2000$	$3000$	$4000$	$5000$
Boiling point ( $^{\circ}$ F)

Choose scales for the axes and graph the equation.
How much does the boiling point decrease when the altitude increases from $1000$ to $3000$ feet? Illustrate this amount on the graph.
At what altitudes is the boiling point less than $204 degrees$ F? Illustrate on the graph.

For each table, choose appropriate scales for the axes and plot the given points.

$x$	$0$	$80$	$90$	$120$
$y$	$6$	$2$	$1.5$	$1$

$x$	$300$	$500$	$800$	$1100$
$y$	$1.2$	$1.3$	$1.5$	$1.9$

$x$	$0.01$	$0.03$	$0.06$	$0.07$
$y$	$minus 0.2$	$minus 1$	$minus 1.1$	$minus 2$

$x$	$0.003$	$0.005$	$0.008$	$0.011$
$y$	$6$	$2$	$1.5$	$1$

For Problems 13–18,

Find the intercepts of the graph.
Graph the equation by the intercept method.

$x + 2 y = 8$

$open parenthesis 8, 0 close parenthesis, open parenthesis 0, 4 close parenthesis$

$2 x - y = 6$

$3 x - 4 y = 12$

$open parenthesis 4, 0 close parenthesis, open parenthesis 0, minus 3 close parenthesis$

$2 x + 6 y = 6$

$\frac{x}{9} - \frac{y}{4} = 1$

$open parenthesis 9, 0 close parenthesis, open parenthesis 0, minus 4 close parenthesis$

$\frac{x}{5} + \frac{y}{8} = 1$

For Problems 19–24,

Find the intercepts of the graph.
Use the intercepts to choose scales for the axes, and then graph the equation by the intercept method.

$20 x = 30 y - 45, 000$

$open parenthesis minus 2250, 0 close parenthesis, open parenthesis 0, 1500 close parenthesis$

$30 x = 45 y + 60, 000$

$0.4 x + 1.2 y = 4.8$

$open parenthesis 12, 0 close parenthesis, open parenthesis 0, 4 close parenthesis$

$3.2 x - 0.8 y = 12.8$

$\frac{2 x}{3} + \frac{3 y}{11} = 1$

$(\frac{3}{2}, 0), (0, \frac{11}{3})$

$\frac{8 x}{7} - \frac{2 y}{7} = 1$

The owner of a gas station has $ $19, 200$ to spend on unleaded gas this month. Regular unleaded costs him $ $2.40$ per gallon, and premium unleaded costs $ $3.20$ per gallon.

How much do $x$ gallons of regular cost? How much do $y$ gallons of premium cost?
Write an equation in general form that relates the amount of regular unleaded gasoline, $x$ , the owner can buy and the amount of premium unleaded, $y$ .
Find the intercepts and sketch the graph.
What do the intercepts tell us about the amount of gasoline the owner can purchase?

$ $2.40 x,$ $ $3.20 y$
$2.40 x + 3.20 y = 19, 200$
The $y$ -intercept, $6000$ gallons, is the amount of premium that the gas station owner can buy if he buys no regular. The $x$ -intercept, $8000$ gallons, is the amount of regular he can buy if he buys no premium.

Five pounds of body fat is equivalent to $16, 000$ calories. Carol can burn $600$ calories per hour bicycling and $400$ calories per hour swimming.

How many calories will Carol burn in $x$ hours of cycling? How many calories will she burn in $y$ hours of swimming?
Write an equation in general form that relates the number of hours, $x$ , of cycling and the number of hours, $y$ , of swimming Carol needs to perform in order to lose $5$ pounds.
Find the intercepts and sketch the graph.
What do the intercepts tell us about Carol's exercise program?

Delbert must increase his daily potassium intake by $1800$ mg. He decides to eat a combination of figs and bananas, which are both low in sodium. There are $9$ mg potassium per gram of fig, and $4$ mg potassium per gram of banana.

How much potassium is in $x$ grams of fig? How much potassium is in $y$ grams of banana?
Write an equation in general form that relates the number of grams, $x$ , of fig and the number of grams, $y$ , of banana Delbert needs to get $1800$ mg of potassium.
Find the intercepts and sketch the graph.
What do the intercepts tell us about Delbert's diet?

$9 x$ mg, $4 y$ mg
$9 x + 4 y = 1800$
The $x$ -intercept, $200$ grams, tells how much fig Delbert should eat if he has no bananas, and the $y$ -intercept, $450$ grams, tells how much banana he should eat if he has no figs.

Leslie plans to invest some money in two CD accounts. The first account pays $3.6 %$ interest per year, and the second account pays $2.8 %$ interest per year. Leslie would like to earn $ $500$ per year on her investment.

If Leslie invests $x$ dollars in the first account, how much interest will she earn? How much interest will she earn if she invests $y$ dollars in the second account?
Write an equation in general form that relates $x$ and $y$ if Leslie earns $$ 500$ interest.
Find the intercepts and sketch the graph.
What do the intercepts tell us about Leslie's investments?

Find the intercepts of the graph for each equation.

$\frac{x}{3} + \frac{y}{5} = 1$
$2 x - 4 y = 1$
$\frac{2 x}{5} - \frac{2 y}{3} = 1$
$\frac{x}{p} + \frac{y}{q} = 1$

e. Why is the equation $\frac{x}{a} + \frac{y}{b} = 1$ called the intercept form for a line?

$open parenthesis 3, 0 close parenthesis, open parenthesis 0, 5 close parenthesis$
$(\frac{1}{2}, 0), (0, \frac{- 1}{4})$
$(\frac{5}{2}, 0), (0, \frac{- 3}{2})$
$open parenthesis p, 0 close parenthesis, open parenthesis 0, q close parenthesis$
The value of $a$ is the $x$ -intercept, and the value of $b$ is the $y$ -intercept.

Write an equation in intercept form (see Problem 29) for the line with the given intercepts. Then write the equation in general form.

$open parenthesis 6, 0 close parenthesis, open parenthesis 0, 2 close parenthesis$
$open parenthesis minus 3, 0 close parenthesis, open parenthesis 0, 8 close parenthesis$
$(\frac{3}{4}, 0), (0, \frac{- 1}{4})$
$open parenthesis v, 0 close parenthesis, open parenthesis 0, minus w close parenthesis$
$(\frac{1}{H}, 0), (0, \frac{1}{T})$

Find the $y$ -intercept of the line $y = m x + b$ .
Find the $x$ -intercept of the line $y = m x + b$ .

$open parenthesis 0, b close parenthesis$
$(\frac{- b}{m}, 0)$ , if $m not equal to 0$

Find the $y$ -intercept of the line $A x + B y = C$ .
Find the $x$ -intercept of the line $A x + B y = C$ .

Write an equation in general form for each line.

$- 2 x + 3 y = 2400$

$3 x + 400 y = 240$

For Problems 37–44,

Solve each equation for $y$ in terms of $x$ . (See the Algebra Skills Refresher Linear Equations and Inequalities to review this skill.)
Graph the equation with your graphing utility in the specified window.
Make a pencil and paper sketch of the graph. Label the scales on your axes, and the coordinates of the intercepts.

$2 + y = 6$

$\begin{aligned} Xmin & = - 10 & Ymin = - 10 \\ Xmax & = 10 & Ymax = 10 \\ Xscl & = 1 & Yscl = 1 \end{aligned}$

$y = 6 - 2 x$

$8 - y + 3 x = 0$

$\begin{aligned} Xmin & = - 10 & Ymin = - 10 \\ Xmax & = 10 & Ymax = 10 \\ Xscl & = 1 & Yscl = 1 \end{aligned}$

$3 x - 4 y = 1200$

$\begin{aligned} Xmin & = - 1000 & Ymin = - 1000 \\ Xmax & = 1000 & Ymax = 1000 \\ Xscl & = 100 & Yscl = 100 \end{aligned}$

$y = \frac{3}{4} x - 300$

$x + 2 y = 500$

$\begin{aligned} Xmin & = - 1000 & Ymin = - 1000 \\ Xmax & = 1000 & Ymax = 1000 \\ Xscl & = 100 & Yscl = 100 \end{aligned}$

$0.2 x + 5 y = 0.1$

$\begin{aligned} Xmin & = - 1 & Ymin = - 0.1 \\ Xmax & = 1 & Ymax = 0.1 \\ Xscl & = 0.1 & Yscl = 0.01 \end{aligned}$

$y = 0.02 - 0.04 x$

$1.2 x - 4.2 y = 3.6$

$\begin{aligned} Xmin & = - 1 & Ymin = - 1 \\ Xmax & = 4 & Ymax = 1 \\ Xscl & = 0.2 & Yscl = 0.1 \end{aligned}$

$70 x + 3 y = y + 420$

$\begin{aligned} Xmin & = 0 & Ymin = 0 \\ Xmax & = 10 & Ymax = 250 \\ Xscl & = 1 & Yscl = 25 \end{aligned}$

$y = 210 - 35 x$

$40 y - 5 x = 780 - 20 y$

$\begin{aligned} Xmin & = - 200 & Ymin = 0 \\ Xmax & = 0 & Ymax = 20 \\ Xscl & = 20 & Yscl = 2 \end{aligned}$

For Problems 45–52,

Find the $x$ - and $y$ -intercepts.
Solve the equation for $y$ .
Choose a graphing window in which both intercepts are visible, and graph the equation with your graphing utility.

$x + 4 y = 100$

$open parenthesis 100, 0 close parenthesis, open parenthesis 0, 25 close parenthesis$
$y = 25 - \frac{1}{4} x$

$2 x - 3 y = - 72$

$25 x - 20 y = 1$

$open parenthesis 0.04, 0 close parenthesis, open parenthesis 0, minus 0.05 close parenthesis$
$y = 1.25 x - 0.05$

$4 x + 75 y = 60, 000$

$\frac{y}{12} - \frac{x}{60} = 1$

$open parenthesis minus 60, 0 close parenthesis, open parenthesis 0, 12 close parenthesis$
$y = 12 + \frac{1}{5} x$

$\frac{x}{80} + \frac{y}{400} = 1$

$- 2 x = 3 y + 84$

$(- 42, 0), (0, - 28)$
$y = \frac{- 2}{3} x - 28$

$7 x = 91 - 13 y$

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.

Sales	Income
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