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Section 3.5 Systems of Equations

Subsection 3.5.1 Overview

In Section 3.5, we expand our focus. So far in Chapter 3, we have analyzed and solved linear functions and used regression to model linear functions that arise from data. We now consider how two or more linear functions interact with each other. As consumers, we are often faced with choices. In this lesson, we will solve systems of equations in order to determine the best deal for our needs.

Much of the mathematics work we have done this semester has depended on making connections among representations. To study and solve systems of equations, the interrelationships between tables, graphs, and equations are critical. Think about the contexts, and create tables, graphs, and equations to extend what you have learned about linear functions to work with them two or more at a time.

Student Page 3.5.2 Linear Functions and Fitness Plans

By now, you are proficient at finding equations. In the student page, Linear Functions and Fitness Plans, you will use tables to locate an interval over which two or more of the fitness plans cost the same amount. You will refine your search for an intersection point with graphs and then by solving pairs of equations.

1.

Consider payment plans for three different fitness centers:

  • Shape Up charges an initiation fee of $39 plus $35 per month per person.

  • Fitness Fun charges an initiation fee of $75 plus $20 per month per person.

  • Be Your Best has no initiation fee and charges $52 per month per person.

Complete Table 3.5.2.1 showing the total cost for the first 6 months for each fitness company's payment plan (initiation fees are only charged once).

Table 3.5.2.1. Fitness Costs
  Amount Paid so far (in dollars)
Month Shape Up Fitness Fun Be Your Best
1 74    
2 109    
3      
4      
5      
6      
\(m\)      

2.

For each set of data, graph (month, amount paid so far). Label each graph with the company's name.

3.

(a)

Which company charges the most after 6 months of membership? How do you know?

(b)

How can you tell from a graph?

(c)

How can you tell from the table?

4.

Which company has the steepest graph? Why does that make sense?

5.

What does the \(y\)-intercept represent for each fitness payment plan?

6.

Find an equation that fits each data set. Include the equation in the last row of Table 3.5.2.1.

7.

You have $500 to spend on a fitness plan this year. Show your work.

(a)

Which fitness plans can you afford?

(b)

For how many months can you afford each fitness plan?

8.

In the summer, you are very active outside. You only want to join a fitness center for 7, 8, or 9 months of the year. The fitness centers offer the same amenities with one exception, Be Your Best also has a pool.

(a)

Determine the cost for all three fitness plans for these numbers of months. Fill in Table 3.5.2.2

Table 3.5.2.2. Fitness Costs for Select Lengths
  Amount Paid so far ($)
Month Shape Up Fitness Fun Be Your Best
7      
8      
9      
(b)

Which fitness plan would you choose and for how many months? Explain.

11.

Consider the following pairs of fitness plans. For what number of months will you pay the same amount for a membership in either plan?

(a)

Shape Up and Fitness Fun:

(b)

Fitness Fun and Be Your Best:

(c)

Be Your Best and Shape Up:

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.

Table 3.5.3.1. Bike Rental Information
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.

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.

(d)

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

6.

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

Homework 3.5.4 Homework

1.

Study the lines in Figure 3.5.4.1.

Figure 3.5.4.1. Lines for Exercise 3.5.4.1
(a)

Estimate the intersection point of the lines.

(b)

Find at least two grid points on each graph. Do not estimate. Label the points on the graph.

(c)

Find an equation for each line. This is called a system of equations in two unknowns because the values of \(x\) and \(y\) of the intersection point are both unknown.

(d)

How can you solve the system of equations algebraically?

(e)

Solve the system algebraically.

2.

Graph the following equations.

\begin{align*} y \amp = 2x - 3\\ y \amp = -0.5x + 4 \end{align*}
(a)

What do you know about the intersection point of the two lines?

(b)

Estimate the intersection point graphically.

(c)

How does the answer to Task 3.5.4.2.a help you determine a way to solve the problem algebraically?

(d)

Solve the problem algebraically.

(e)

Replace the variables in both equations with the coordinates of the intersection point. Does the point you found satisfy both equations? Should it? Explain.

3.

A video game is available for your cell phone in two versions. The free versions charges $0.99 for each ‚Äúbooster,‚ÄĚ an award that helps you win a level more easily. The full version costs $5.99 then charges $0.49 per booster.

(a)

How much will the game cost you to play overall based on the number of boosters you buy? Fill in Table 3.5.4.2 Include equations in the last column for both versions of the game.

Table 3.5.4.2. Video Game Booster Costs
Number of Boosters 0 1 2 3 4 5 \(x\)
Free Version              
Full Version              
(b)

For each of the video games, complete the table with values accurate to at least 2 decimal places. Do not estimate from a graph!

Table 3.5.4.3. More Information On Video Game Costs
  Free Version Full Version
\(y\)-intercept    
slope of the line    
equation that fits the line    
\(x\)-intercept    
(c)

Graph the equations. Estimate the coordinates of the intersection point. Write the intersection point as an ordered pair \((x, y)\text{.}\)

(d)

For what number of boosters will both versions of the game cost the same amount? Solve the problem algebraically and say how you know you are correct.

(e)

What is the domain for the Free Version? Explain.

(f)

What is the range for the Free Version? Explain.

4.

(a)

What is the domain of each fitness plan? How do you know?

(b)

What is the domain of each of the functions you found without context?

(c)

How are the domains different?

(d)

How are they related?

5.

(a)

What is the range of each fitness plan? How do you know?

(b)

What is the range of each of the functions you found without context?

(c)

How are the ranges different?

(d)

How are they related?

6.

You've studied \(y\)-intercepts throughout Chapter 2 and Chapter 3. In Section 3.5, you were asked to find \(x\)-intercepts without a definition.

(a)

What is an \(x\)-intercept?

(b)

How can you find an \(x\)-intercept from a graph?

(c)

How can you find an \(x\)-intercept in a table?

(d)

How can you find an \(x\)-intercept from an equation?

7.

Suppose you owe your parents $2000 for a car they bought you. You're paying it off at a rate of $100 each month.

(a)

Write an equation to model this situation.

(b)

What does the slope mean in this situation?

(c)

What does the \(y\)-intercept mean in this situation?

(d)

What does the \(x\)-intercept mean in this situation?

(e)

Find the \(x\)-intercept.

8.

Revisit Systems of Equations in Context: Renting Bicycles Solve each problem. Explain!

(a)

Based on pricing alone, which bike rental company would you not use? Why?

(b)

Which bike rental company will you use if you want to rent bikes for less than 4 hours?

(c)

Which bike rental company will you use if you want to rent bikes for more than 4 hours?

9.

You want to rent a game system and some games to try the system out before you buy it. You check with several companies and find the following information for one-week rentals:

  • Game Galaxy charges $50 for the game system rental and $5 per game.

  • Electronic Emporium charges $25 for the game system rental and $10 per game.

  • Why Buy? doesn't charge for the game system rental, but charges $15 per game.

(a)

Complete Table 3.5.4.4.

Table 3.5.4.4. Game System Fees
Rental Company Game System
Rental Fee
Rental Per Game Equation
Game Galaxy      
Electronic Emporium      
Why Buy?      
(b)

Graph the equations. Estimate where each pair of lines intersects. Write the solutions as ordered pairs in Table 3.5.4.5.

Table 3.5.4.5. Game System Intersections
Pair of Companies Intersection Point
Game Galaxy and Electronic Emporium  
Game Galaxy and Why Buy?  
Electronic Emporium and Why Buy?  
(c)

Which game system rental company gives the best deal if who want to rent a small number of games (1 or 2 games)? How do you know? Show your work.

(d)

Which game system rental company gives the best deal for people who want to rent more than 10 games? How do you know? Show your work.

(e)

Do any of the game system rental companies ever charge the same amount for the same number of games rented? (Note: Include the game system rental fee.)

If so, which companies and for what number of games? How much do they charge for this number of games?

(f)

Solve the system of equations algebraically for one pair of companies. List the companies. Show your work.