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
[latex]C=0.10x+50[/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.Answer: Let [latex]x[/latex] equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as [latex]x+17[/latex]. The sum of the two numbers is 31. We usually interpret the word is as an equal sign.
[latex]\begin{array}{rl}x+\left(x+17\right)&=31\hfill \\ 2x+17&=31\hfill&\text{Simplify and solve}.\hfill \\ 2x&=14\hfill \\ x&=7\hfill\end{array}[/latex]
The second number would then be [latex]x+17=7+17=24[/latex]
The two numbers are [latex]7[/latex] and [latex]24[/latex].
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?
Answer:
- The model for Company A can be written as [latex]A=0.05x+34[/latex]. This includes the variable cost of [latex]0.05x[/latex] plus the monthly service charge of $34. Company B’s package charges a higher monthly fee of [latex]$40[/latex], but a lower variable cost of [latex]0.04x[/latex]. Company B’s model can be written as [latex]B=0.04x+\$40[/latex].
- If the average number of minutes used each month is [latex]1,160[/latex], we have the following:
[latex]\begin{array}{l}\text{Company }A\hfill&=0.05\left(1,160\right)+34\hfill \\ \hfill&=58+34\hfill \\ \hfill&=92\hfill \\ \hfill \\ \text{Company }B\hfill&=0.04\left(1,160\right)+40\hfill \\ \hfill&=46.4+40\hfill \\ \hfill&=86.4\hfill \end{array}[/latex]So, Company B offers the lower monthly cost of [latex]$86.40[/latex] as compared with the [latex]$92[/latex] monthly cost offered by Company A when the average number of minutes used each month is [latex]1,160[/latex].
- If the average number of minutes used each month is [latex]420[/latex], we have the following:
[latex]\begin{array}{l}\text{Company }A\hfill&=0.05\left(420\right)+34\hfill \\ \hfill&=21+34\hfill \\ \hfill&=55\hfill \\ \hfill \\ \text{Company }B\hfill&=0.04\left(420\right)+40\hfill \\ \hfill&=16.8+40\hfill \\ \hfill&=56.8\hfill \end{array}[/latex]If the average number of minutes used each month is [latex]420[/latex], then Company A offers a lower monthly cost of $55 compared to Company B’s monthly cost of [latex]$56.80[/latex].
- 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 [latex]\left(x,y\right)[/latex] 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.
[latex]\begin{array}{l}0.05x+34=0.04x+40\hfill \\ 0.01x=6\hfill \\ x=600\hfill \end{array}[/latex]Check the x-value in each equation.[latex]\begin{array}{l}0.05\left(600\right)+34=64\hfill \\ 0.04\left(600\right)+40=64\hfill \end{array}[/latex]Therefore, a monthly average of [latex]600[/latex] talk-time minutes renders the plans equal.
A graph representing the relationship between cell phone plan cost and minutes used.
Licenses & Attributions
CC licensed content, Original
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Write and Solve Linear Equation - Number Problem with Given Sum. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Write Linear Equations to Model and Compare Cell Phone Plans with Data Usage. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- College Algebra. Provided by: OpenStax Authored by: Abramson, Jay, et al.. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download for free at : http://cnx.org/contents/[email protected]:1/Preface.