Lesson 2 Elvira’s Equations Solidify Understanding

Jump Start

Quantities are attributes that can be counted or measured, such as a dozen eggs, $25 miles$ , or $60 minutes$ . Notice that quantities always include a unit of measure: eggs, miles, minutes. Some quantities are derived from combining other measurements, such as miles per gallon, or feet per minute.

1.

Describe how you would measure the fuel efficiency of your family car in miles per gallon.

2.

Describe how you would measure the speed of a baseball in feet per second.

3.

Describe how you would measure the area of a rectangular room in square feet.

4.

Describe how you would calculate the revenue collected by a roadside fruit stand if they sell $20$ bags of oranges for $$ 5$ per bag.

Learning Focus

Use units to interpret and solve equations that contain primarily variables that represent quantities, such as a formula.

What does it mean to solve an equation that contains multiple variables?

How can I use descriptions and units associated with variables to guide my thinking about solving such equations?

Open Up the Math: Launch, Explore, Discuss

Elvira, the cafeteria manager, likes to keep track of the things she can count or measure in the cafeteria. She hopes this will help her improve the efficiency of the cafeteria. To remind herself to keep track of important quantities, she has made a table of variables and descriptions of the things she wants to record. Here is a table of things she has decided to keep track of.

1.

Determine the units that fit in the third column of the table for each of the variables whose descriptions are given.

Symbol	Meaning (description of what the symbol means in context)	Units (what is counted or measured)
$S$	Number of students who buy lunch in the salad line
$W$	Number of students who buy lunch in the sandwich line
$P$	Number of students who buy lunch in the pizza line
$F$	Number of food servers in the cafeteria
$M_{T}$	Number of minutes it takes to serve lunch to all students
$C$	Number of classes in the school
$P_{L}$	Price per lunch
$A$
$R$
$T$
$D_{F}$
$M$

Elvira has written the following equation to describe a cafeteria relationship that seems meaningful to her. She has introduced a new variable, $A$ , to describe this relationship.

$A = \frac{S + W + P}{C}$

2.

What does $A$ represent in terms of the cafeteria? Record this information in the table.

3.

Using what you know about manipulating equations, solve this equation for $S$ . Your solution will be of the form $S = an expression$ written in terms of the variables $A$ , $C$ , $W$ , and $P$ .

4.

Does your expression for $S$ make sense in terms of the meanings of the other variables? Explain why or why not.

Here is another one of Elvira’s equations.

$R = P_{L} (S + W + P)$

5.

What does $R$ represent in terms of the cafeteria? Record this information in the table.

6.

Using what you know about manipulating equations, solve this equation for $P_{L}$ .

7.

Does your expression for $P_{L}$ make sense in terms of the meanings of the other variables? Explain why or why not.

8.

Elvira notices that she uses the expression $S + W + P$ a lot in writing other expressions. She decides to represent this expression using the variable $T$ so that $T = S + W + P$ . What does $T$ represent in terms of the cafeteria? Record this information in the table.

Elvira is having a meeting with the staff members who work in the lunchroom. She has created a couple of new equations for the food servers.

$D_{F} = \frac{T \cdot P_{L}}{F}$

$M = \frac{M_{T}}{T}$

9.

a.

What does $D_{F}$ represent in terms of the cafeteria? Record this information in the table.

b.

Solve this equation for $P_{L}$ . Describe why your solution makes sense in terms of the other variables.

Pause and Reflect

10.

$M = \frac{M_{T}}{T}$

a.

What does $M$ represent in terms of the cafeteria? Record this information in the table.

b.

Solve this equation for $T$ . Describe why your solution makes sense in terms of the other variables.

11.

One of the staff members suggests that they need to write expressions for each of the following sentences. Using the variables in the table, what would these expressions look like?

a.

The average number of students served each minute.

b.

The average number of minutes students wait in the pizza line.

Ready for More?

Use the variables in the table to create other meaningful expressions. What might be the longest expression (in terms of number of variables required) that can be written using these variables?

Takeaways

Equations that involve multiple variables are called literal equations. If the variables in the equation represent meaningful quantities, literal equations are also referred to as formulas.

To solve a literal equation for one of its variables, I can

If the variables in the equation represent meaningful quantities, I can

When determining the meaning of an expression that contains multiple quantities, I need to attend to the units and operations. Some strategies for doing so include

Vocabulary

Lesson Summary

In this lesson, we learned how to solve literal equations for one of its variables using inverse operations with both variables and numbers. Literal equations are formulas for describing the relationships between multiple quantities. Interpreting the meaning of expressions involving quantities in terms of their units can be a tool for checking our algebraic work while solving equations.

Retrieval

Solve each of the equations for the indicated variable. Explain what you did to solve.

1.

Solve for $r$ : $d = r t$

2.

Solve for $w$ : $l = w + 9$

3.

Solve for $n$ : $\frac{1}{3} n = b$

4.

Find the domain and range for the function represented in the graph.

5.

Graph the inequality on the number line.

$x > - 3$