Int-Alg Chapter Summary and Review

Section 1.6 Chapter Summary and Review

Subsection 1.6.1 Glossary

mathematical model
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linear model
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evaluate an expressions
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solve an equation
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increasing graph
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decreasing graph
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inequality
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intercepts
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linear equation
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solution of an equation
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graph of an equation
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ordered pair
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rate
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rate of change
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net change
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slope
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slope-intercept form
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point-slope form
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Subsection 1.6.2 Key Concepts

A mathematical model is a simplified description of reality that helps us understand a system or process.
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We can describe a relationship between variables with a table of values, a graph, or an equation.
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Linear models have equations of the form:

$\begin{matrix} y = (starting value) + (rate of change) \cdot x \end{matrix}$

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The general form for a linear equation is: $A x + B y = C .$
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We can use the intercepts to graph a line. The intercepts are also useful for interpreting a model.
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The graph of an equation in two variables is just a picture of all its solutions.
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Lines have constant slope.
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The slope of a line gives us the rate of change of one variable with respect to another.
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Formulas for Linear Models
- slope: $m = \frac{Δ y}{Δ x} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}},$ $x_{1} \neq x_{2}$
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- slope-intercept form: $y = b + m x$
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- point-slopet form: $y = y_{1} + m (x - x_{1})$
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The slope-intercept form is useful when we know the initial value and the rate of change.
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The point-slope form is useful when we know the rate of change and one point on the line.
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Exercises 1.6.3 Chapter 1 Review Problems

1.

Last year, Pinwheel Industries introduced a new model calculator. It cost $2000 to develop the calculator and $20 to manufacture each one.

Complete the table of values showing the total cost, $C,$ of producing $n$ calculators.
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$n$ $100$ $500$ $800$ $1200$ $1500$

$C$

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Write an equation that expresses $C$ in terms of $n .$
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Graph the equation by hand.
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$grid$

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What is the cost of producing 1000 calculators? Illustrate this as a point on your graph.
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How many calculators can be produced for $10,000? Illustrate this as a point on your graph.
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2.

The world’s oil reserves were 2100 billion barrels in 2005; total annual consumption is 28 billion barrels per year.

Complete the table of values that shows the remaining oil reserves $R$ in terms of time $t$ (in years since 2005).
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$t$ $5$ $10$ $15$ $20$ $25$

$R$

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Write an equation that expresses $R$ in terms of $t .$
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Find the intercepts and graph the equation by hand.
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$grid$

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What do the intercepts tell us about the world’s oil supply?
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3.

Alida plans to spend part of her vacation in Atlantic City and part in Saint-Tropez. She estimates that after airfare her vacation will cost $60 per day in Atlantic City and $100 per day in Saint-Tropez. She has $1200 to spend after airfare.

Write an equation that relates the number of days, $C,$ Alida can spend in Atlantic City and the number of days, $T,$ in Saint-Tropez.
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Find the intercepts and graph the equation by hand.
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$grid$

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If Alida spends 10 days in Atlantic City, how long can she spend in Saint-Tropez?
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What do the intercepts tell us about Alida’s vacation?
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Exercise Group.

For Problems 4–9, graph the equation on graph paper. Use the most convenient method for each problem.

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4.

\frac{x}{6} - \frac{y}{12} = 1

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5.

50 x = 40 y - 20, 000

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6.

1.4 x + 2.1 y = 8.4

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7.

3 x - 4 y = 0

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8.

3 x - 4 y = 0

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9.

4 x = - 12

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10.

The table shows the amount of oil,

B

(in thousands of barrels), left in a tanker

t

minutes after it hits an iceberg and springs a leak.

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$t$	$0$	$10$	$20$	$30$
$B$	$800$	$750$	$700$	$650$

Write a linear function for $B$ in terms of $t .$
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Choose appropriate window settings on your calculator and graph your function.
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Give the slope of the graph, including units, and explain the meaning of the slope in terms of the oil leak.
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11.

An interior decorator bases her fee on the cost of a remodeling job. The accompanying table shows her fee,

F,

for jobs of various costs,

C,

both given in dollars.

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$C$	$5000$	$10, 000$	$20, 000$	$50, 000$
$F$	$1000$	$1500$	$2500$	$5500$

Write a linear function for $F$ in terms of $C .$
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Choose appropriate window settings on your calculator and graph your function.
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Give the slope of the graph, including units, and explain the meaning of the slope in terms of the the decorator’s fee.
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Exercise Group.

For Problems 12–15, find the slope of the line segment joining the points.

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12.

(- 1, 4), (3, - 2)

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13.

(5, 0), (2, - 6)

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14.

(6.2, 1.4), (- 2.3, 4, 8)

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15.

(0, - 6.4), (- 5.6, 3.4)

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Exercise Group.

For Problems 16–17, which tables could describe variables related by a linear equation?

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16.

$r$ $E$

$1$ $5$

$2$ $\frac{5}{2}$

$3$ $\frac{5}{3}$

$4$ $\frac{5}{4}$

$5$ $1$

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$s$ $t$

$10$ $6.2$

$20$ $9.7$

$30$ $12.6$

$40$ $15.8$

$50$ $19.0$

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17.

$w$ $A$

$2$ $- 13$

$4$ $- 23$

$6$ $- 33$

$8$ $- 43$

$10$ $- 53$

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$x$ $G$

$0$ $0$

$2$ $5$

$4$ $10$

$8$ $20$

$16$ $40$

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Exercise Group.

For Problems 18–19, the table gives values for a linear equation in two variables. Fill in the missing values.

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18.

$d$	$V$
$- 5$	$- 4.8$
$- 2$	$- 3$
	$- 1.2$
$6$	$1.8$
$10$

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19.

$q$	$S$
$- 8$	$- 8$
$- 4$	$36$
$3$
	$200$
$9$	$264$

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20.

The planners at AquaWorld want the small water slide to have a slope of 25%. If the slide is 20 feet tall, how far should the end of the slide be from the base of the ladder?

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Exercise Group.

For Problems 21–24, find the slope and

y

-intercept of the line.

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21.

2 x - 4 y = 5

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22.

\frac{1}{2} x + \frac{2}{3} y = \frac{5}{6}

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23.

8.4 x + 2.1 y = 6.3

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24.

y - 3 = 0

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Exercise Group.

For Problems 25 and 26,

Graph by hand the line that passes through the given point with the given slope.
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Find the equation of the line.
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25.

(- 4, 6);

m = \frac{- 2}{3}

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26.

(2, - 5);

m = \frac{3}{2}

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27.

The rate at which air temperature decreases with altitude is called the lapse rate. In the troposphere, the layer of atmosphere that extends from the Earth’s surface to a height of about 7 miles, the lapse rate is about

{3.6}^{\circ} F

for every 1000 feet. (Source: Ahrens, 1998)

If the temperature on the ground is $62^{\circ} F,$ write an equation for the temperature, $T,$ at an altitude of $h$ feet.
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What is the temperature outside an aircraft flying at an altitude of 30,000 feet? How much colder is that than the ground temperature?
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What is the temperature at the top of the troposphere?
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Exercise Group.

For Problems 28 and 29, find an equation for the line passing through the two given points.

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28.

(3, - 5),

(- 2, 4)

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29.

(0, 8),

(4, - 2)

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30.

The population of Maple Rapids was 4800 in 2005 and had grown to 6780 by 2020. Assume that the population increases at a constant rate.

Make a table of values showing two data points.
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Find a linear equation that expresses the population, $P,$ of Maple Rapids in terms of the number of years, $t,$ since 2005.
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State the slope of the line, including units. What does it tell us about the population?
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Exercise Group.

For Problems 31 and 32,

Find the slope and $y$ -intercept of the line.
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Write an equation for the line.
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31.

$line$

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32.

$line$

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33.

What is the slope of the line whose intercepts are

(- 5, 0)

and

(0, 3) ?

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34.

Find the $x$ - and $y$ -intercepts of the line $\frac{x}{4} - \frac{y}{6} = 1 .$
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What is the slope of the line in part (a)?
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35.

Find the $x$ - and $y$ -intercepts of the line $y = 2 + \frac{3}{2} (x - 4) .$
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Find the point on the line whose $x$ -coordinate is 4. Can there be more than one such point?
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36.

Find an equation in slope-intercept form for the line of slope

\frac{6}{5}

that passes through

(- 3, - 4) .

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$r$	$E$
$1$	$5$
$2$	$\frac{5}{2}$
$3$	$\frac{5}{3}$
$4$	$\frac{5}{4}$
$5$	$1$

$n$	$100$	$500$	$800$	$1200$	$1500$
$C$

$s$	$t$
$10$	$6.2$
$20$	$9.7$
$30$	$12.6$
$40$	$15.8$
$50$	$19.0$

$w$	$A$
$2$	$- 13$
$4$	$- 23$
$6$	$- 33$
$8$	$- 43$
$10$	$- 53$