### 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 [latex]$0.10/mi[/latex] in addition to a flat rate. In this case, a known cost, such as [latex]$0.10/mi[/latex], is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write [latex]0.10x[/latex]. 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 [latex]$0.10/mi[/latex] plus a daily fee of [latex]$50[/latex]. We can use these quantities to model an equation that can be used to find the daily car rental cost [latex]C[/latex].

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 |
[latex]x,\text{ }x+a[/latex] |

Twice a number | [latex]2x[/latex] |

One number is a more than another number |
[latex]x,\text{ }x+a[/latex] |

One number is a less than twice another number |
[latex]x,2x-a[/latex] |

The product of a number and a, decreased by b |
[latex]ax-b[/latex] |

The quotient of a number and the number plus a is three times the number |
[latex]\dfrac{x}{x+a}\normalsize =3x[/latex] |

The product of three times a number and the number decreased by b is c |
[latex]3x\left(x-b\right)=c[/latex] |

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

- Identify known quantities.
- Assign a variable to represent the unknown quantity.
- If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
- Write an equation interpreting the words as mathematical operations.
- 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 [latex]17[/latex] and their sum is [latex]31[/latex]. 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 [latex]$34[/latex] plus [latex]$.05/min[/latex] talk-time. Company B charges a monthly service fee of [latex]$40[/latex] plus [latex]$.04/min[/latex] talk-time.

- Write a linear equation that models the packages offered by both companies.
- If the average number of minutes used each month is [latex]1,160[/latex], which company offers the better plan?
- If the average number of minutes used each month is [latex]420[/latex], which company offers the better plan?
- 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.