# What Is an Equation?

An equation is a mathematical expression with a left-hand side and a right-hand side, separated by an equal sign in the middle. The equal sign indicates that the left-hand side is the same as the right-hand side. The goal of an equation is to find the value of a variable. To do this, you have to tidy up such that the variable ends up alone on one side with the numbers on the other side. Now you’ll learn the techniques needed to tidy up and solve different equations.

## Change Side, Change Sign

You might remember that equations have to be balanced. When you do something on one side of the equal sign, you have to do the same thing on the other side. Let’s look at some examples.

Example 1

$\begin{array}{llll}\hfill x+3& =4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x+3-3& =4-3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =4-3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ You want to get rid off 3 on the left-hand side. If you subtract 3 on the left-hand side, the number 3 disappears. The 3 you removed from the left-hand side must also be subtracted from the right-hand side. Then the 3 appears on the other side of the equal sign, but with the opposite sign.

Example 2

$\begin{array}{llll}\hfill x-2& =5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x-2+2& =5+2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =5+2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$ You want to get rid of $-2$ on the left-hand side. If you add 2 on the left-hand side, the number 2 disappears. The 2 you removed from the left-hand side must also be added to the right-hand side. Then the 2 appears on the other side of the equal sign, but with the opposite sign.

Rule

### ChangeSide,ChangeSign

When adding or subtracting a term on both sides, you move the term over to the other side and change the sign. Therefore, I call this the “change side, change sign” rule.