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.

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.

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

Answer:

1. 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].
2. 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].
3. 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].
4. 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.