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 106.

Find three consecutive odd integers so that the sum of twice the first, the second and three times the third is $152$ .

First $x$	Make the first $x$
Second $x +2$	Odd numbers so we add $2$ (same as even!)
Third $x +4$	Add $2$ more ( $4$ total)to get the third
$2F + S + 3T = 152$	Twice the first gives $2F$ and three times the third gives $3T$
$2(x)+ (x +2) +3(x +4) = 152$	Replace $F$ , $S$ , and $T$ with what we labeled them
$2x + x +2 +3x + 12 = 152$	Distribute through parenthesis
$6x + 14 = 152$	Combine like terms $2x + x +3x$ and $2+ 14$
$\underline {− 14 − 14}$	Subtract $14$ from both sides
$\underline {6x = 138}$	Variable is multiplied by $6$
$6 \quad \quad6$	Divide both sides by $6$
$x = 23$	Our solution for $x$
First $23$	Replace $x$ with $23$ in the original list
Second $(23)+2 = 25$ Third $(23)+4 = 27$	The numbers are $23$ , $25$ , and $27$

When we started with our first, second, and third numbers for both even and odd we had $x$ , $x + 2$ , and $x + 4$ . The numbers added do not change with odd or even, it is our answer for $x$ that will be odd or even.

Another example of translating English sentences to mathematical sentences comes from geometry. A well known property of triangles is that all three angles will always add to $180$ . For example, the first angle may be $50$ degrees, the second $30$ degrees, and the third $100$ degrees. If you add these together, $50 + 30 + 100 = 180$ . We can use this property to find angles of triangles.

World View Note: German mathematician Bernhart Thibaut in 1809 tried to prove that the angles of a triangle add to 180 without using Euclid’s parallel postulate (a point of much debate in math history). He created a proof, but it was later shown to have an error in the proof.