1.8.3 Writing Equations and Isolating Variables

Example 1

Suppose we want to find $a$ when $b$ is ${20}^{\circ }$.

$a+a+b=180$

$2a+20=180$

$2a=180-20$

$2a=160$

$a=80$

Example 2

What is the value of $b$ if $a$ is ${72.5}^{\circ }$?

$a+a+b=180$

$72.5+72.5+b=80$

$145+b=180$

$b=35$

Notice that when $b$ is given, we did the same calculation repeatedly to find $a$: We substituted $b$ into the first equation, subtracted $b$ from 180, and then divided the result by 2.

Instead of taking these steps over and over whenever we know $b$ and want to find $a$, we can rearrange the equation to isolate $a$:

$a+a+b=180$

$2a+b=180$

$2a=180-b$

$a=\frac{180-b}{2}$

$a=90-\frac{1}{2}b$

This equation is equivalent to the first one. To find $a$, we can now simply substitute any value of $b$ into this equation and evaluate the expression on the right side.

Likewise, we can write an equivalent equation to make it easier to find $b$ when we know $a$:

$a+a+b=180$

$2a+b=180$

$b=180-2a$

Rearranging an equation to isolate one variable is called solving for a variable. In this example, we have solved for $a$ and for $b$. All three equations are equivalent. Depending on what information we have and what we are interested in, we can choose a particular equation to use.