### Student Page 3.5.3 Systems of Equations in Context: Renting Bicycles

Four different bike rental companies are competing for your business on Mackinac Island. Which one gives the best deal for the number of hours you plan to rent a bike?

This activity helps you determine the best deal. It also helps you make sense of the importance of domain and range both in context and without a context.

When you finish the Systems of Equations in Context: Renting Bicycles activity, answer the following questions for Section 3.5:

How can you solve a system of two linear equations in two unknowns graphically?

How can you also solve a system algebraically?

You and your friends plan to rent bikes while visiting Mackinac Island.

Island Rentals charges a base fee of $16 plus $6 per hour per bike.

Bikes Unlimited does not charge a base fee, but charges $10 per hour per bike.

#### 1.

##### (a)

You plan to use bikes for 3 hours. Which company is least expensive?

##### (b)

Which company should you use if you plan to use bikes for 6 hours?

##### (c)

Tegan noticed that if the group uses bikes for exactly 4 hours, the companies charge the same for rentals. Is she correct? Why or why not?

#### 2.

##### (a)

Find the slope, both intercepts, and meanings of all of these for each company.

##### (b)

Find equations for both bike rental companies. Enter them in Table 3.5.3.1.

Rental Company | Slope | \(y\)-intercept | \(x\)-intercept | Equation |
---|---|---|---|---|

Island Rentals | ||||

Bikes Unlimited | ||||

Beckmann's Bikes | ||||

Colleen's Cyclery |

##### (c)

Graph the equations. Label axes and scales.

##### (d)

Plot the data from Student Page Exercise 3.5.3.1 on each graph. Do your equations fit the data you found in Student Page Exercise 3.5.3.1?

#### 3.

Recall that the domain of a function is the set of all possible values of the independent variable (\(x\) or whatever variable you are using for the context) for which the function is defined. The range is the set of possible values of the dependent variable (usually \(y\text{,}\) sometimes expressed as \(f(x)\)) that result from using the same function.

##### (a)

Find the domain in terms of the bike rental context.

##### (b)

Find the domain disregarding the context.

##### (c)

How are these similar? How are they different?

##### (d)

Find the range in terms of one of the bike rental companies.

##### (e)

Find the range disregarding the context.

##### (f)

How are these similar? How are they different?

#### 4.

Suppose there are two other bike rental companies on the island.

Beckmann's Bikes charges a base fee of $5 and $10 per hour per bike.

Colleen's Cyclery charges a base fee of $20 and $5 per hour per bike.

##### (a)

How are the charges from these bike rental companies similar to the charges for Island Rentals and Bikes Unlimited?

##### (b)

Find equations for each company. Record slopes, intercepts, and equations in the table.

##### (c)

Graph the equations on the same coordinate plane you used in Student Page Exercise 3.5.3.2.

##### (d)

Compare the graphs of these two companies with those above. What do you notice?

#### 5.

##### (a)

Are any of the lines in Student Page Exercise 3.5.3.2 or Student Page Exercise 3.5.3.4 parallel? If so, which ones? How do you know?

##### (b)

In the context of bike rentals, what aspects of the bike rental make the lines modeling the rental charges parallel?

#### 6.

Write and answer two questions related to bike rentals and the fees the four companies above charge.