Int-Alg Intercepts

Write an equation for the amount of gasoline, $g,$ left in the tank after Rosa has driven for $d$ miles.
Find the intercepts of the graph. What do they tell us about the problem situation?

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Hint: Rewrite the sentence with mathematical symbols:

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(gasoline left) = (gallons in full tank) - (mileage rate) \times (miles driven)

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Answer.

$g = 11 - \frac{1}{12} d$
$(0, 11) :$ The tank has 11 gallons before Rosa begins driving.

$(528, 0) :$ After Rosa drives 528 miles, the tank will be empty.

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Subsection 1.3.3 General Form for a Linear Equation

The order of the terms in a linear equation does not matter. For example, the equation in Example 1.1.4 of Section 1.1

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C = 5 + 3 t can be written equivalently as - 3 t + C = 5

and the equation in Example 1.1.11 of that section,

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g = 20 - \frac{1}{12} d can be written  as \frac{1}{12} d + g = 20

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This form of a linear equation,

A x + B y = C,

is called the general form.

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Definition 1.3.7. General Form for a Linear Equation.

The general form for a linear equation is

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A x + B y = C

(where

A

and

B

cannot both be 0).

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Checkpoint 1.3.8. QuickCheck 2.

What is the general form of a linear equation?

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$y = m x + b$
$A x + B y = C$
Any equation whose graph is a straight line
Set $x = 0$ and solve for $y .$

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Some linear models are easier to use when they are written in the general form.

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Example 1.3.9.

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.

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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.

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Solution.

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 $x = 10$ 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.
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Checkpoint 1.3.10. Practice 3.

The manager at Breadbasket Bakery has $120 to spend on advertising. An ad in the local newspaper costs $15, and posters cost $4 each. She decides to buy

x

ads and

y

posters. Write an equation relating

x

and

y .

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Hint: Use the general form for a linear equation. What is the total amount of money the manager will spend?

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Answer.

15 x + 4 y = 120

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Subsection 1.3.4 Intercept Method of Graphing

Because we really need only two points to graph a linear equation, we might as well find the intercepts and use them to draw the graph.

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To Graph a Linear Equation by the Intercept Method.

Find the horizontal and vertical intercepts.
Plot the intercepts, and draw the line through the two points.

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Example 1.3.11.

Find the $x$ - and $y$ -intercepts of the graph of $150 x + 50 y = 3000 .$
Use the intercepts to graph the equation. $[TK]$

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Solution.

To find the $x$ -intercept, we set $y = 0 .$

$\begin{aligned} 150 x - 50 (0) & = 3000 & Simpify. \\ 150 x & = 3000 & Divide both sides by 150. \\ x & = 20 \end{aligned}$

The $x$ -intercept is the point $(20, 0) .$ To find the $y$ -intercept, we set $x = 0 .$

$\begin{aligned} 150 (0) - 50 y & = 3000 & Simpify. \\ 506 & = 3000 & Divide both sides by 50. \\ y & = 60 \end{aligned}$

The $y$ -intercept is the point $(0, 60) .$
We scale both axes in intervals of 10 and then plot the two intercepts, $(20, 0)$ and $(0, 60) .$ We draw the line through them, as shown.
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$line with positive intercepts$

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Checkpoint 1.3.12. QuickCheck 3.

Describe the intercept method of graphing a linear equation.

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Make a table of values and plot the points.
Extend the line until it crosses both axes.
Solve for $y$ in terms of $x .$
Plot the points where $x = 0$ and where $y = 0,$ then draw the line through them

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Checkpoint 1.3.13. Practice 4.

Find the

x

- and

y

-intercepts of the equation in Practice 3 (about the Breadbasket Bakery), and use the intercepts to graph the equation.

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Hint: Choose convenient scales for the

x

- and

y

-axes.

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Answer.

(0, 30), (8, 0)

$line with positive intercepts$

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Subsection 1.3.5 Two Forms for Linear Equations

We have now seen two forms for linear equations: the general linear form,

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A x + B y = C

and the form for a linear model,

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y = (starting value) + (rate) \times t

Sometimes it is useful to convert an equation from one form to the other.

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Example 1.3.14.

Write the equation $4 x - 3 y = 6$ in the form for a linear model.
Write the equation $y = - 9 - \frac{3}{2} x$ in general linear form.

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Solution.

We would like to solve for $y$ in terms of $x .$ We first isolate the $y$ -term on one side of the equation.

$\begin{aligned} 4 x - 3 y & = 6 & Subtract 4 x from both sides. \\ - 3 y & = 6 - 4 x & Divide both sides by - 3 . \\ \frac{- 3 y}{- 3} & = \frac{6 - 4 x}{- 3} & Simplify: divide each term by - 3 . \\ y & - 2 + \frac{4}{3} x \end{aligned}$
Write the equation in the form $A x + B y = C .$

$\begin{aligned} y & = - 9 - \frac{3}{2} x & Add \frac{3}{2} x to both sides. \\ \frac{3}{2} x + y & = - 9 \end{aligned}$

We can write the equation with integer coefficients by clearing the fractions. We multiply both sides of the equation by $2 .$

$\begin{aligned} 2 (\frac{3}{2} x + y) & = - 9 (2) \\ 3 x + 2 y & = - 18 \end{aligned}$

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Caution 1.3.15.

Do not confuse solving for

y

in terms of

x

with finding the

y

-intercept. Compare:

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In the Example above, we solved $4 x - 3 y = 6$ for $y$ in terms of $x$ to get

$y = - 2 + \frac{4}{3} x .$

This is still an equation in two variables; it is just another (equivalent) form of the original equation.
To find the $y$ -intercept of the same equation, we first set $x = 0,$ then solve for $y,$ as follows:

$\begin{aligned} 4 (0) - 3 y & = 6 \\ y = - 2 \end{aligned}$

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This gives us a particular point on the graph, namely,

(0, - 2);

the point whose

x

-coordinate is 0.

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Checkpoint 1.3.16. Practice 5.

Write the equation $150 x + 50 y = 3000$ in the form for a linear model.
Write the equation $y = 0.15 x - 3.8$ in general linear form with integer coefficients.

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Answer.

$y = - 3 x + 60$
$- 15 x + 100 y = - 380$ or $3 x - 20 y = 76$

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Exercises 1.3.6 Problem Set 1.3

Warm Up

1.

The owner of a movie theater needs to bring in $1000 revenue at each screening in order to stay in business. He sells adults’ tickets for $5 each and children’s tickets at $2 each.

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How much revenue does he earn from selling $x$ adults’ tickets?
How much revenue does he earn from selling $y$ children’s tickets?
Write an equation in $x$ and $y$ for the number of tickets he must sell at each screening

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2.

Karel needs 45 milliliters of a 40% solution of carbolic acid. He plans to mix some 20% solution with some 50% solution.

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How much carbolic acid is in $x$ milliliters of the 20% solution?
How much carbolic acid is in $y$ milliliters of the 50% solution?
How much carbolic acid is in the solution Karel needs?
Write an equation in $x$ and $y$ for the amount of each solution Karel should mix.

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Exercise Group.

For Problems 3 and 4, solve the equation for

y

in terms of

x .

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3.

3 x + 5 y = 16

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4.

20 x = 30 y - 45, 000

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Skills Practice

Exercise Group.

For Problems 5-8,

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

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5.

9 x - 12 y = 36

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6.

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

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7.

4 y = 20 + 2.5 x

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8.

30 x = 45 y + 60, 000

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9.

Find the intercepts of the graph for each equation.

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

e. Why is the equation

\frac{x}{a} + \frac{y}{b} = 1

called the intercept form for a line?

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10.

Write an equation in intercept form (see Problem 9) for the line with the given intercepts. Then write the equation in general form.

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

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11.

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

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12.

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

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Exercise Group.

For Problems 13-16, write an equation in general form for the line.

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Exercise Group.

For Problems 17-20, write the equation in two standard forms:

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the general linear form, $A x + B y = C,$ with integer coefficients, and
the form for a linear model, $y = (starting value) + (rate) \times x$

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17.

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

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18.

\frac{8 x}{7} + \frac{2 y}{7} = 1

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19.

0.4 x = 4.8 - 1.2 y

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20.

- 0.8 y = 12.8 - 3.2 x

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Applications

21.

Delbert must increase his daily potassium intake by 1800 mg. He decides to eat a combination of figs and bananas. One gram of fig contains 9 mg of potassium, and one gram of banana contains 4 mg of potassium.

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How many mg of potassium are in $x$ grams of figs?
How many mg of potassium are in $y$ grams of bananas?
Write an equation for the number of grams of fig, $x,$ and the number of grams of banana, $y,$ that Delbert needs to eat daily.
Find the intercepts of the graph. What do the intercepts tell us about Delbert’s diet?

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22.

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

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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 that relates the number of hours, $x,$ of cycling and $y,$ of swimming Carol needs to perform in order to lose 5 pounds.
Find the intercepts of the graph. What do the intercepts tell us about Carol’s exercise program?

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23.

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

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Complete the table of values for the diver’s altitude $h$ after $t$ minutes. (A depth of $400$ feet is the same as an altitude of $- 400$ feet.)

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

$h$
Write an equation for the diver’s altitude, $h,$ in terms of the number of minutes, $t,$ elapsed.
Find the intercepts and sketch the graph.
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$t$ $h$

$0$

$0$

$grid$
Explain what each intercept tells us about this problem.

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24.

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)

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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.

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

$y$
Find the intercepts of the graph.
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$x$ $y$

$0$

$0$

$grid$
Use the intercepts to help you choose appropriate scales for the axes, and graph the equation.
What do the intercepts tell us about the problem?
What does the point $(288, 100)$ mean in this context?

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25.

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.

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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?

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26.

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.

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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?

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$x$	$8$	$10$	$12$	$14$
$y$	$36$	$30$	$24$	$18$

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

$t$	$h$
$0$
	$0$

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