Example

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: write a Linear Equation in the form of to Solve a Real-World Application

1. Write a linear equation that models the packages offered by both companies.
2. If the average number of minutes used each month is 1,160, which company offers the better plan?
3. If the average number of minutes used each month is 420, 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 $40, 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 1,160, 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 $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.
3. If the average number of minutes used each month is 420, 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 420, then Company A offers a lower monthly cost of $55 compared to Company B’s monthly cost of $56.80.
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 600 talk-time minutes renders the plans equal.

example: write a linear equation in one variable to model and solve an application

Answer: Let [latex]G[/latex] represent the number of green marbles in the bag and [latex]B[/latex] represent the number of blue marbles. We have that [latex]G + B = 77[/latex]. But we also have that there are 17 more green marbles than blue. That is, the number of green marbles is the same as the number of blue marbles plus 17. We can translate that as [latex]G = B+17[/latex]. Since we've found a way to express the variable [latex]G[/latex] in terms of [latex]B[/latex], we can write one equation in one variable. [latex]B+17+B=77[/latex], which we can solve for [latex]B[/latex]. [latex-display]2B=60[/latex], so [latex]B=30[/latex-display] There are [latex]30[/latex] Blue marbles in the bag and [latex]30+17 = 47[/latex] is the number of green marbles.