# 2.3 Models and applications  (Page 2/9)

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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 $\text{\hspace{0.17em}}36,$ find the numbers.

11 and 25

## 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.

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?
1. The model for Company A can be written as $\text{\hspace{0.17em}}A=0.05x+34.\text{\hspace{0.17em}}$ This includes the variable cost of $\text{\hspace{0.17em}}0.05x\text{\hspace{0.17em}}$ plus the monthly service charge of $34. Company B ’s package charges a higher monthly fee of$40, but a lower variable cost of $\text{\hspace{0.17em}}0.04x.\text{\hspace{0.17em}}$ Company B ’s model can be written as $\text{\hspace{0.17em}}B=0.04x+\text{}40.$
2. 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.

3. 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.

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 $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ 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.

$\begin{array}{ccc}\hfill 0.05x+34& =& 0.04x+40\hfill \\ \hfill 0.01x& =& 6\hfill \\ \hfill x& =& 600\hfill \end{array}$

Check the x- value in each equation.

$\begin{array}{ccc}\hfill 0.05\left(600\right)+34& =& 64\hfill \\ \hfill 0.04\left(600\right)+40& =& 64\hfill \end{array}$

Therefore, a monthly average of 600 talk-time minutes renders the plans equal. See [link]

Find a linear equation to model this real-world application: It costs ABC electronics company $2.50 per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of$350 for utilities and \$3,300 for salaries. What are the company’s monthly expenses?

$C=2.5x+3,650$

## Using a formula to solve a real-world application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area    of a rectangular region, $\text{\hspace{0.17em}}A=LW;$ the perimeter    of a rectangle, $\text{\hspace{0.17em}}P=2L+2W;$ and the volume    of a rectangular solid, $\text{\hspace{0.17em}}V=LWH.\text{\hspace{0.17em}}$ When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

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