Lesson 1 Business Cents Solidify Understanding

Learning Focus

Solve systems of linear equations that represent a context.

What strategies might I use to find the point of intersection of two linear equations?

What does each strategy reveal? What might each strategy conceal?

How do I use the solution to a system of equations to make decisions within a context?

Technology guidance for today’s lesson:

Equation Solving: Casio ClassPad
Equation Solving Using a Table: Casio fx-9750GIII
Solve Systems of Equations by Graphing: Casio ClassPad Casio fx-9750GIII

Open Up the Math: Launch, Explore, Discuss

The Martinez twins, Carlos and Clarita, are trying to find a way to make money during summer vacation. When they overhear their aunt complaining about how difficult it is to find someone to care for her pets while she will be away on a trip, Carlos and Clarita know they have found the perfect solution. Not only do they have a large, unused storage shed on their property where they can house animals, they also have a spacious fenced backyard where the pets can play.

Carlos and Clarita need to clean the storage shed where they plan to board the pets. They have decided to hire a company to clean the windows. After collecting the following information, they have come to you for help deciding which window cleaning company they should hire.

Sunshine Express Window Cleaners charges $$ 50$ for each service call, plus $$ 10$ per window.
“Pane”less Window Cleaners charges $$ 25$ for each service call, plus $$ 15$ per window.

1.

Which company would you recommend, and why? Prepare an argument to convince Carlos and Clarita that your recommendation is reasonable. (It is always more convincing if you can support your claim in multiple ways. How might you support your recommendation using a table? A graph? Algebra?)

Once the windows are clean, Carlos and Clarita also want to have the window frames repainted to make the storage shed look more appealing to potential customers. They have decided to hire a company to paint the windows. Again, they need your help deciding which painter they should hire.

Paint by Numbers charges $$ 100$ for sanding the frames and masking the windows with tape, and $$ 16$ per hour for painting, since different employees do different parts of the work.

Paint it Right charges $$ 24$ per hour for all labor, plus a $$ 38$ waste disposal fee.

2.

Which company would you recommend, and why? Prepare an argument to convince Carlos and Clarita that your recommendation is reasonable. (Support your claim in multiple ways including a table, a graph, and algebra.)

Both paint companies allow the customer to purchase their own paint supplies. Carlos has done some research and found the following information:

Handyman Tools and Supplies sells a small painter’s kit that contains $2$ paint brushes and $4$ rolls of masking tape for $$ 16$ . They also sell a large painter’s kit that contains $5$ paint brushes and $15$ rolls of masking tape for $$ 50$ .

Carlos’s dad is planning to paint a couple of rooms in their house, and has offered to buy any left-over supplies. Carlos is trying to figure out how much he would charge his dad for each paint brush and each roll of tape. To keep track of the information, he has written the following equations: $\begin{matrix} 2 p + 4 r = 16 \\ 5 p + 15 r = 50 \end{matrix}$ Clarita is looking at Carlos’s notes and asks, “What do $p$ and $r$ stand for?”

3.

How would you answer Clarita’s question?

Clarita says, “I think I can find the values of $p$ and $r$ using a table or a graph.”

Carlos says, “I think I can find the values of $p$ and $r$ using algebra.”

4.

Solve for $p$ and $r$ using both a graphical strategy and an algebraic strategy.

Ready for More?

While solving the system of equations about the painting supplies, what assumptions did you make about the context? Are these assumptions necessarily true?

Takeaways

While each equation in a system of equations has many solutions, the solution to a system of linear equations is:

Systems of equations can be solved using tables, graphs or algebra. The solution to the system is revealed in different ways, and each strategy has its own pros and cons.

Using a graph:

How the solution is revealed:

Pros:

Cons:

Using a table:

How the solution is revealed:

Pros:

Cons:

Using algebra:

How the solution is revealed:

Pros:

Cons:

There are two methods for solving a system of linear equations algebraically by substitution:

Method 1:

Method 2:

These methods work equally well for systems in which the linear equations are linear functions, and for systems in which the linear equations are linear combinations with a constant sum.

A linear combination is:

Vocabulary

Lesson Summary

In this lesson, we reviewed strategies for solving systems of linear equations, including using tables, graphs, and algebra. We examined two different algebraic methods, both of which involve substitution of expressions for variables. These methods work with systems of linear equations that are linear functions, as well as systems of linear equations that are linear combinations.

Retrieval

Create a table and graph for the given equations. Then state the $x$ -intercept and $y$ -intercept and circle them on the table and graph.

1.

$y = - 2 x + 8$

$x$	$y$

2.

$x + y = 7$

$x$	$y$

3.

$2 (x - 3) + 5 \geq 3 x - 10$