### Your Turn

5.1

5.3

5.5

5.7

5.8

5.10

5.11

5.12

5.13

5.14

5.15

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

1.

12 = - 1, which is false; therefore, this is a false statement, and the equation has no solution.

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

5.23

5.24

5.25

5.26

5.27

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

5.44

5.45

5.47

5.48

5.49

5.50

5.51

5.52

5.53

5.54

5.55

5.56

5.57

5.58

5.59

5.60

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( {-3,3} \right),\left( {-2,2} \right),\left( {-1,0} \right),\left( {0,-1} \right),\left( {2,-2} \right),\left( {4,-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

5.72

5.73

5.74

5.75

5.76

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

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

5.92

5.93

5.94

5.95

1.

\left\{ \begin{array}{l}240h + 160C \ge 800\\1.40h + 0.50C \le 5\\h \ge 0\\C \ge 0\end{array} \right.

3.

The point (3, 2) is not in the solution region. Omar would not choose to eat 3 hamburgers and 2 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 \leq 20
and 3a + 5b \leq 70.
The summary is: p = 4a + 6b, a + b \leq 20, and 3a + 5b \leq 70.

5.99

1.

The constraints are:

15 \leq x \leq 22

13 \leq y \leq 19

So the system is:

T = 20x + 28y

15 \leq x \leq 22

13 \leq y \leq 19

15 \leq x \leq 22

13 \leq y \leq 19

So the system is:

T = 20x + 28y

15 \leq x \leq 22

13 \leq y \leq 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)}&{ = }&{6\left( {38 + 10} \right)?}\\{8\left( {36} \right)}&{ = }&{6\left( {48} \right)?}\\{288}&{ = }&{288 ✓}\\\end{array}

Let

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

10.

It is a correct solution strategy.

Let

\begin{array}{rcl}{x}&{ = }&{- \,2}\\{7 + 4\left( {2 + 5\left( { - \,2} \right)} \right)}&{ \mathop = \limits^? }&{3\left( {6\left( { -\,2} \right) + 7} \right)-\left( {13\left( { -\,2} \right) + 36} \right)}\\{7 + 4\left( {2-10} \right)}&{ \mathop = \limits^? }&{3\left( { - 12 + 7} \right)-\left( { -\,26 + 36} \right) }\\{7 + 4\left( { -\,8} \right)}&{ \mathop = \limits^? }&{3\left( { - 5} \right)-\left( {10} \right) }\\{7-32}&{ \mathop = \limits^? }&{ -\,15-10}\\{-\,25}&{ = }&{-\,25✓}\end{array}

Let

\begin{array}{rcl}{x}&{ = }&{- \,2}\\{7 + 4\left( {2 + 5\left( { - \,2} \right)} \right)}&{ \mathop = \limits^? }&{3\left( {6\left( { -\,2} \right) + 7} \right)-\left( {13\left( { -\,2} \right) + 36} \right)}\\{7 + 4\left( {2-10} \right)}&{ \mathop = \limits^? }&{3\left( { - 12 + 7} \right)-\left( { -\,26 + 36} \right) }\\{7 + 4\left( { -\,8} \right)}&{ \mathop = \limits^? }&{3\left( { - 5} \right)-\left( {10} \right) }\\{7-32}&{ \mathop = \limits^? }&{ -\,15-10}\\{-\,25}&{ = }&{-\,25✓}\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!