Setting up a Linear Equation to Solve a Real-World Application
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 us use the car rental example above. 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 . 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 .
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 | |
Twice a number | |
One number is a more than another number | |
One number is a less than twice another number | |
The product of a number and a, decreased by b | |
The quotient of a number and the number plus a is three times the number | |
The product of three times a number and the number decreased by b is c |
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 the solution can be explained in words, including the units of measure.
Example 1: Modeling a Linear Equation to Solve an Unknown Number Problem
Find a linear equation to solve for the following unknown quantities: One number exceeds another number by and their sum is . Find the two numbers.Solution
Let equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as . The sum of the two numbers is 31. We usually interpret the word is as an equal sign.
The two numbers are and .
Try It 1
Find a linear equation to solve for the following unknown quantities: One number is three more than twice another number. If the sum of the two numbers is , find the numbers. SolutionExample 2: Setting Up a Linear Equation to Solve a Real-World Application
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.- Write a linear equation that models the packages offered by both companies.
- If the average number of minutes used each month is 1,160, which company offers the better plan?
- If the average number of minutes used each month is 420, which company offers the better plan?
- How many minutes of talk-time would yield equal monthly statements from both companies?
Solution
- The model for Company A can be written as . This includes the variable cost of plus the monthly service charge of $34. Company B’s package charges a higher monthly fee of $40, but a lower variable cost of . Company B’s model can be written as .
- If the average number of minutes used each month is 1,160, we have the following:
So, Company B offers the lower monthly cost of $86.40 as compared with the $92 monthly cost offered by Company A when the average number of minutes used each month is 1,160.
- If the average number of minutes used each month is 420, we have the following:
If the average number of minutes used each month is 420, then Company A offers a lower monthly cost of $55 compared to Company B’s monthly cost of $56.80.
- To answer the question of how many talk-time minutes would yield the same bill from both companies, we should think about the problem in terms of coordinates: At what point are both the x-value and the y-value equal? We can find this point by setting the equations equal to each other and solving for x.
Check the x-value in each equation.Therefore, a monthly average of 600 talk-time minutes renders the plans equal.
Figure 2