Applications of Linear Equations

This section this textbook explains how to translate the situations described in word problems to equations and provides a variety of examples. Read the chapter and work through the problems. Some examples involved the geometric facts you have learned in Unit 2.

Example 104.

The sum of three consecutive integers is $93$ . What are the integers?

First $x$	Make the first number $x$ .
Second $x +1$	To get the next number we go up one or $+1$
Third $x +2$	Add another $1$ ( $2$ total) to get the third
$F + S + T = 93$	First $(F)$ plus Second $(S)$ plus Third $(T)$ equals $93$
$(x)+(x + 1)+ (x +2) = 93$	Replace $F$ with $x$ , $S$ with $x +1$ , and $T$ with $x +2$
$x + x +1+ x +2 = 93$	Here the parenthesis aren't needed.
$3x +3 = 93$	Combine like terms $x + x + x$ and $2 + 1$
$\underline {− 3 \quad − 3}$	Add $3$ to both sides
$\underline {3x = 90}$	The variable is multiplied by $3$
$3 \quad \quad 3$	Divide both sides by $3$
$x = 30$	Our solution for $x$
First $30$	Replace $x$ in our original list with $30$
Second $(30) +1 = 31$ Third $(30) +2 = 32$	The numbers are $30$ , $31$ , and $32$

Sometimes we will work consecutive even or odd integers, rather than just consecutive integers. When we had consecutive integers, we only had to add $1$ to get to the next number so we had $x$ , $x + 1$ , and $x + 2$ for our first, second, and third number respectively. With even or odd numbers they are spaced apart by two. So if we want three consecutive even numbers, if the first is $x$ , the next number would be $x + 2$ , then finally add two more to get the third, $x + 4$ . The same is true for consecutive odd numbers, if the first is $x$ , the next will be $x + 2$ , and the third would be $x + 4$ . It is important to note that we are still adding $2$ and $4$ even when the numbers are odd. This is because the phrase "odd" is referring to our $x$ , not to what is added to the numbers. Consider the next two examples.