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Study Guides > College Algebra

Write a Linear Equation to Solve an Application

Learning Objectives

  • Write a linear equation to express the relationship between unknown quantities
  • Write a linear equation that models two different cell phone packages
  • Use a linear model to answer questions
To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write 0.10x0.10x. This expression represents a variable cost because it changes according to the number of miles driven. If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost CC.
C=0.10x+50C=0.10x+50
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 x, x+ax,\text{ }x+a
Twice a number 2x2x
One number is a more than another number x, x+ax,\text{ }x+a
One number is a less than twice another number x,2x−ax,2x-a
The product of a number and a, decreased by b ax−bax-b
The quotient of a number and the number plus a is three times the number xx+a=3x\frac{x}{x+a}=3x
The product of three times a number and the number decreased by b is c 3x(x−b)=c3x\left(x-b\right)=c

How To: Given a real-world problem, model a linear equation to fit it.

  1. Identify known quantities.
  2. Assign a variable to represent the unknown quantity.
  3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
  4. Write an equation interpreting the words as mathematical operations.
  5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.

Example: Modeling a Linear Equation to Solve an Unknown Number Problem

Find a linear equation to solve for the following unknown quantities: One number exceeds another number by 1717 and their sum is 3131. Find the two numbers.

Answer: Let xx equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as x+17x+17. The sum of the two numbers is 31. We usually interpret the word is as an equal sign.

x+(x+17)=312x+17=31Simplify and solve.2x=14x=7x+17=7+17=24\begin{array}{l}x+\left(x+17\right)\hfill&=31\hfill \\ 2x+17\hfill&=31\hfill&\text{Simplify and solve}.\hfill \\ 2x\hfill&=14\hfill \\ x\hfill&=7\hfill \\ \hfill \\ x+17\hfill&=7+17\hfill \\ \hfill&=24\hfill \end{array}
The two numbers are 77 and 2424.

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 3636, find the numbers.

Answer: 11 and 25

Use Desmos to graph the two models you created. Then, determine the number of minutes for which either plan would cost the same amount. You will need to adjust the graph settings to do this, and the following tutorial shows you how. https://youtu.be/En_PkyA-4_4

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