# 2.3 Models and applications

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In this section you will:
• Set up a linear equation to solve a real-world application.
• Use a formula to solve a real-world application. Credit: Kevin Dooley

Josh is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum of points that can be earned is 100. Is it possible for Josh to end the course with an A? A simple linear equation will give Josh his answer.

Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

## Setting up a linear equation to solve a real-world application

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 $\text{\hspace{0.17em}}0.10x.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}C.$

$C=0.10x+50$

When dealing with real-world applications, there are certain expressions that we can translate directly into math. [link] lists some common verbal expressions and their equivalent mathematical expressions.

Verbal Translation to Math Operations
One number exceeds another by a $x,\text{​}\text{\hspace{0.17em}}x+a$
Twice a number $2x$
One number is a more than another number $x,\text{​}\text{\hspace{0.17em}}x+a$
One number is a less than twice another number $x,\text{\hspace{0.17em}}2x-a$
The product of a number and a , decreased by b $ax-b$
The quotient of a number and the number plus a is three times the number $\frac{x}{x+a}=3x$
The product of three times a number and the number decreased by b is c $3x\left(x-b\right)=c$

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.

## 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 $\text{\hspace{0.17em}}17\text{\hspace{0.17em}}$ and their sum is $\text{\hspace{0.17em}}31.\text{\hspace{0.17em}}$ Find the two numbers.

Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as $\text{\hspace{0.17em}}x+17.\text{\hspace{0.17em}}$ The sum of the two numbers is 31. We usually interpret the word is as an equal sign.

The two numbers are $\text{\hspace{0.17em}}7\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}24.$

#### Questions & Answers

prove sin²x+cos²x=3+cos4x
Kiddy Reply
the difference between two signed numbers is -8.if the minued is 5,what is the subtrahend
jeramie Reply
the difference between two signed numbers is -8.if the minuend is 5.what is the subtrahend
jeramie
what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
ask
Ans
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
what are examples of natural number
sani Reply
an equation for the line that goes through the point (-1,12) and has a slope of 2,3
Katheryn Reply
3y=-9x+25
Ishaq
show that the set of natural numberdoes not from agroup with addition or multiplication butit forms aseni group with respect toaaddition as well as multiplication
Komal Reply
x^20+x^15+x^10+x^5/x^2+1
Urmila Reply
evaluate each algebraic expression. 2x+×_2 if ×=5
Sarch Reply
if the ratio of the root of ax+bx+c =0, show that (m+1)^2 ac =b^2m
Awe Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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