# Writing a Linear Equation to Solve an Application

### Learning Outcomes

- Write a linear equation to express the relationship between unknown quantities.
- Write a linear equation to model a real-world situation.
- Write a linear equation in one variable to solve problems with two unknowns.

### Recall expressions and equations

Recall that a mathematical**expression**consists of

**terms**connected by addition or subtraction, each term of which consists of

**variables**and numbers connected by multiplication or division. A number that multiplies a variable, such as the [latex]2[/latex] in the term [latex]2x[/latex] is called the

**coefficient**of the variable. An

**equation**is a mathematical statement of the equivalency of two expressions, one on either side of an equal sign.

**Expressions**may be combined or simplified by rearranging or doing operations on the terms.

**Equations**may be solved for the value or values of the variable that make the statement of equivalency true.

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]\Large\frac{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 situation, write a linear equation to model it

- Identify known and unknown 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 in the problem as mathematical operations.
- Solve the equation, check to be sure your answer is reasonable, and give the answer using the language and units of the original situation.

### 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.

### Try It

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 [latex]36[/latex], find the numbers.Answer: 11 and 25

[ohm_question]142770-142775[/ohm_question]### Recall the forms of equations of lines

**Slope-intercept form**: [latex]y=mx+b[/latex], where [latex]x \text{and } y[/latex] represent the coordinates of any point on the line, [latex]m[/latex] represents the slope of the line, and [latex]b[/latex] represents the initial value, or the y-intercept. You can solve for [latex]b[/latex] by substituting a known point for [latex]x \text{ and } y[/latex] and the slope of the line for [latex]m[/latex]. The

**point-slope form**, [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex] simplifies to slope-intercept form when solved for [latex]y[/latex]. Substitute the coordinates of a point for [latex]{y}_1 \text{ and } {x}_1[/latex] and the slope for [latex]m[/latex] then simplify.

### Example: write a Linear Equation in the form of 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.- Write a linear equation that models the packages offered by both companies.
- If the average number of minutes used each month is 1,160, which company offers the better plan?
- If the average number of minutes used each month is 420, 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 $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]. - 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. - 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. - 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.

### try it

There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $0.04/min talk-time. Company B charges a monthly service fee of $45 plus $0.03/min talk-time. Use [latex]x[/latex] for your variable. a) Write an equation that models the monthly cost for company A. b) Write an equation that models the monthly cost for company B. C) If the average number of minutes used each month is 1,162, how much is the monthly cost for each company?Answer: a) [latex]A=0.04x+34[/latex] b) [latex]B = 0.03x+45[/latex] c) Company A's cost would be $80.48, and Company B's cost would be $79.86

[ohm_question]92426[/ohm_question]### Try It

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?Answer: [latex] C=2.5x+3,650[/latex]

[ohm_question]93001[/ohm_question]### example: write a linear equation in one variable to model and solve an application

A bag is filled with green and blue marbles. There are 77 marbles in the bag. If there are 17 more green marbles than blue marbles, find the number of green marbles and the number of blue marbles in the bag. How many marbles of each color are in the bag?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.

### Try it

A cash register contains only five dollar and ten dollar bills. It contains twice as many five's as ten's and the total amount of money in the cash register is 560 dollars. How many ten's are in the cash register?Answer: There are 28 ten dollar bills in the cash register.

[ohm_question]13829[/ohm_question]## Licenses & Attributions

### CC licensed content, Original

- Revision and Adaptation.
**Provided by:**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://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites.**License:**CC BY: Attribution.**License terms:**Download for free at http://cnx.org/contents/[email protected]. - Learn Desmos: Change Graph Settings.
**Authored by:**Desmos.**License:**All Rights Reserved.**License terms:**Standard YouTube Licesnse. - Question ID 142770, 142775.
**Authored by:**Alyson Day.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC- BY + GPL. - Question ID 13829.
**Authored by:**James Sousa.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC- BY + GPL. - Question ID 92426, 93001.
**Authored by:**Michael Jenck.**License:**CC BY: Attribution.**License terms:**IMathAS Community License CC- BY + GPL. - Writing and Solving Linear Equations.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution. - Solve a Coin Problem Using an Equation in One Variable.
**Authored by:**James Sousa (Mathispower4u.com).**License:**CC BY: Attribution.