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
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 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 and their sum is . Find the two numbers.Answer: 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 second number would then be
The two numbers are and .
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.- Write a linear equation that models the packages offered by both companies.
- If the average number of minutes used each month is , which company offers the better plan?
- If the average number of minutes used each month is , which company offers the better plan?
- How many minutes of talk-time would yield equal monthly statements from both companies?
Answer:
- 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 , 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 .
- If the average number of minutes used each month is , we have the following:
If the average number of minutes used each month is , 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 talk-time minutes renders the plans equal.
