## Set Up a Linear Equation to Solve an Application

### Learning Outcomes

• Translate words into algebraic expressions and equations
• Define a process for solving word problems

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let use an example of a car rental company. The company charges $0.10/mi$ in addition to a flat rate. In this case, a known cost, such as $0.10/mi$, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write $0.10x$. This expression represents a variable cost because it changes according to the number of miles driven.

If a quantity is independent of a variable, we usually just add or subtract it according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges  $0.10/mi$ plus a daily fee of $50$. We can use these quantities to model an equation that can be used to find the daily car rental cost $C$.

$C=0.10x+50$

When dealing with real-world applications, there are certain expressions that we can translate directly into math. The table lists some common verbal expressions and their equivalent mathematical expressions.

Verbal Translation to Math Operations
One number exceeds another by a $x,\text{ }x+a$
Twice a number $2x$
One number is a more than another number $x,\text{ }x+a$
One number is a less than twice another number $x,2x-a$
The product of a number and a, decreased by b $ax-b$
The quotient of a number and the number plus a is three times the number $\dfrac{x}{x+a}\normalsize =3x$
The product of three times a number and the number decreased by b is c $3x\left(x-b\right)=c$

### How To: Given a real-world problem, model a linear equation to fit it

1. Identify known quantities.
2. Assign a variable to represent the unknown quantity.
3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
4. Write an equation interpreting the words as mathematical operations.
5. Solve the equation. Be sure to explain the solution in words including the units of measure.

### Example

Find a linear equation to solve for the following unknown quantities: One number exceeds another number by $17$ and their sum is $31$. Find the two numbers.

In the following video, we show another example of how to translate an expression in English into a mathematical equation that can then be solved.

In the next example we will write equations that will help us compare cell phone plans.

### Example

There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34$ plus $.05/min$ talk-time. Company B charges a monthly service fee of $40$ plus $.04/min$ talk-time.

1. Write a linear equation that models the packages offered by both companies.
2. If the average number of minutes used each month is $1,160$, which company offers the better plan?
3. If the average number of minutes used each month is $420$, which company offers the better plan?
4. How many minutes of talk-time would yield equal monthly statements from both companies?

The following video shows another example of writing two equations that will allow you to compare two different cell phone plans.