### Your Turn

5.1

5.5

5.7

5.8

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5.11

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5.16

1.

Answers will vary. For example: You can rent a paddleboard for $25 per hour with a water shoe purchase of $75. If you spent $200, how many hours did you rent the paddle board for?

You rented the paddle board for 5 hours.

You rented the paddle board for 5 hours.

5.17

5.18

1.

$-12=-12$, which is true; therefore, this is a true statement, and there are infinitely many solutions.

5.19

5.20

5.21

5.22

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5.24

5.25

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5.28

1.

$a$ = 1 U.S. dollar, and $b$ = 1.21 Canadian dollars, the ratio is 1 to 1.21; or 1:1.21; or $\frac{1}{1.21}$ .

5.29

1.

With $a$ = 170 pounds on Earth, and $b$ = 64 pounds on Mars, the ratio is 170 to 64; or 170:64; or $\frac{170}{64}$.

5.30

5.31

5.32

5.33

1.

The constant of proportionality (centimeters divided by inches) is 2.54. This tells you that there are 2.54 centimeters in one inch.

5.34

5.35

5.36

5.37

1.

The scale is $1\text{inch}=91.25\text{miles}$. The other borders would calculate as: eastern and western borders are 273.75 miles, and northern border is 365 miles.

5.38

5.39

5.41

5.42

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5.62

5.64

1.

Both George and Mike have two phone numbers. Each $x$-value is not matched with only one $y$-value. This relation is not a function.

5.66

5.67

5.68

5.69

1.

The ordered pairs of the relation are: $\left\{\left(-3,\phantom{\rule{thinmathspace}{0ex}}3\right),\left(-2,\phantom{\rule{thinmathspace}{0ex}}2\right),\left(-1,\phantom{\rule{thinmathspace}{0ex}}0\right),\left(0,\phantom{\rule{thinmathspace}{0ex}}-1\right),\left(2,\phantom{\rule{thinmathspace}{0ex}}-2\right),\left(4,\phantom{\rule{thinmathspace}{0ex}}-4\right)\right\}$.

5.70

1.

The graph crosses the $x$-axis at the point (2, 0). The $x$-intercept is (2, 0). The graph crosses the $y$-axis at the point (0, −2). The $y$-intercept is (0, −2).

5.71

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5.77

5.78

1.

(0, 20) is the $y\text{-intercept}$ and represents that there were 20 teachers at Jones High School in 1990. There is no $x\text{-intercept}$.

2.

In the first 5 years the slope is 2; this means that on average, the school gained 2 teachers every year between 1990 and 1995. Between 1995 and 2000, the slope is 4; on average the school gained 4 teachers every year. Then the slope is 0 between 2000 and 2005 meaning the number of teachers remained the same. There was a decrease in teachers between 2005 and 2010, represented by a slope of –2. Finally, the slope is 4 between 2010 and 2020, which indicates that on average the school gained 4 teachers every year.

3.

Answers will vary. Jones High School was founded in 1990 and hired 2 teachers per year until 1995, when they had an increase in students and they hired 4 teachers per year for the next 5 years. Then there was a hiring freeze, and no teachers were hired between 2000 and 2005. After the hiring freeze, the student population decreased, and they lost 2 teachers per year until 2010. Another surge in student population meant Jones High School hired 4 new teachers per year until 2020 when they had 80 teachers at the school.

5.79

3.

The slope, 2, means that the height $h$ increases 2 inches when the shoe size(s) increases 1 size.

5.80

3.

The slope, 4, means that the weekly cost, $C$, increases by $4 when the number of pizzas sold, $p$, increases by 1. The $C$-intercept means that when the number of pizzas sold is 0, the weekly cost is $25.

5.83

5.84

5.85

5.86

5.87

5.88

1.

Jenna burns 8.3 calories per minute circuit training and 11.2 calories per minute while on the elliptical trainer.

5.90

5.91

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5.94

5.95

3.

The point $(3,2)$ is not in the solution region. Omar would not choose to eat 3 hamburgers and 2 cookies.

4.

The point $(2,4)$ is in the solution region. Omar might choose to eat 2 hamburgers and 4 cookies.

5.96

1.

With $a=$ the number of bags of apples sold, and $b=$ the number of bunches of bananas sold, the objective function is $P=4a+6b$.

5.97

5.98

1.

The constraints are $a+b\le 20$
and $3a+5b\le 70$.
The summary is: $P=4a+6b$, $a+b\le 20$, and $3a+5b\le 70$.

5.99

1.

The constraints are:

$15\le x\le 22$

$13\le y\le 19$

So the system is:

$T=20x+28y$

$15\le x\le 22$

$13\le y\le 19$

$15\le x\le 22$

$13\le y\le 19$

So the system is:

$T=20x+28y$

$15\le x\le 22$

$13\le y\le 19$

5.100

1.

The maximum value for the profit $P$ occurs when $x=15$ and $y=5$. This means that to maximize their profit, the Robotics Club should sell 15 bags of apples and 5 bunches of bananas every day.

### Check Your Understanding

9.

It is a correct solution strategy.

Let

$\begin{array}{rcl}x& =& 38\\ 8\left(38-2\right)& \stackrel{?}{=}& 6\left(38+10\right)\\ 8\left(36\right)& \stackrel{?}{=}& 6\left(48\right)\\ 288& =& 288\u2713\end{array}$

Let

$\begin{array}{rcl}x& =& 38\\ 8\left(38-2\right)& \stackrel{?}{=}& 6\left(38+10\right)\\ 8\left(36\right)& \stackrel{?}{=}& 6\left(48\right)\\ 288& =& 288\u2713\end{array}$

10.

It is a correct solution strategy.

Let

$\begin{array}{rcl}x& =& -\phantom{\rule{thinmathspace}{0ex}}2\\ 7+4\left(2+5\left(-\phantom{\rule{thinmathspace}{0ex}}2\right)\right)& \stackrel{?}{=}& 3\left(6\left(-\phantom{\rule{thinmathspace}{0ex}}2\right)+7\right)-\left(13\left(-\phantom{\rule{thinmathspace}{0ex}}2\right)+36\right)\\ 7+4\left(2-10\right)& \stackrel{?}{=}& 3\left(-12+7\right)-\left(-\phantom{\rule{thinmathspace}{0ex}}26+36\right)\\ 7+4\left(-\phantom{\rule{thinmathspace}{0ex}}8\right)& \stackrel{?}{=}& 3\left(-5\right)-\left(10\right)\\ 7-32& \stackrel{?}{=}& -\phantom{\rule{thinmathspace}{0ex}}15-10\\ -\phantom{\rule{thinmathspace}{0ex}}25& =& -\phantom{\rule{thinmathspace}{0ex}}25\u2713\end{array}$

Let

$\begin{array}{rcl}x& =& -\phantom{\rule{thinmathspace}{0ex}}2\\ 7+4\left(2+5\left(-\phantom{\rule{thinmathspace}{0ex}}2\right)\right)& \stackrel{?}{=}& 3\left(6\left(-\phantom{\rule{thinmathspace}{0ex}}2\right)+7\right)-\left(13\left(-\phantom{\rule{thinmathspace}{0ex}}2\right)+36\right)\\ 7+4\left(2-10\right)& \stackrel{?}{=}& 3\left(-12+7\right)-\left(-\phantom{\rule{thinmathspace}{0ex}}26+36\right)\\ 7+4\left(-\phantom{\rule{thinmathspace}{0ex}}8\right)& \stackrel{?}{=}& 3\left(-5\right)-\left(10\right)\\ 7-32& \stackrel{?}{=}& -\phantom{\rule{thinmathspace}{0ex}}15-10\\ -\phantom{\rule{thinmathspace}{0ex}}25& =& -\phantom{\rule{thinmathspace}{0ex}}25\u2713\end{array}$

11.

This is not a correct solution strategy. The negative sign is not distributed correctly in the second line of the solution strategy. The second line should read $8x+7-2x+9=22-4x+4$.

16.

The Enjoyable Cab Company, because the cab fare will be $ 0.20 less than what it would cost to take a taxi from the Nice Cab Company.

17.

Luis is; there are infinitely many solutions. If this is solved using the general strategy, it simplifies to $0=0$. This is a true statement, so there are infinitely many solutions.

41.

Yes he can, but barely. At 37 miles per gallon, Albert can drive $499.5$ miles. While in theory he can make it, he probably should fill up with gasoline somewhere along the way!