An equation is a mathematical expression with a left-hand side and a right-hand side, separated by an equals sign ($=$) in the middle. The equals sign indicates that the left-hand side is the same as the right-hand side. You can use an equation is to find the value of a variable. To do this, you have to tidy up each side of the equation, such that the variable ends up alone on one side, with the numbers (known as constants) on the other side.

Here, you’ll learn the techniques needed to tidy up and solve different equations.

You might have learned that equations have to be balanced. This means that when you do something on one side of the equal sign, you have to do the same thing on the other side, to keep the balance. Let’s look at some examples.

Example 1

$$\begin{array}{llll}\hfill x{+3}& =4\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x+3-3& =4-3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =4{-3}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =1\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ With this equation, you want to get rid of the 3 on the left-hand side, so that the variable $x$ can be left on its own. If you subtract $3$ on the left-hand side, the constant term $+3$ disappears. The $3$ you removed from the left-hand side must also be subtracted from the right-hand side to keep the equation in balance. Then, the $3$ appears on the other side of the equal sign, but with the opposite sign.

Example 2

$$\begin{array}{llll}\hfill x{-2}& =5\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x-2+2& =5+2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =5{+2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =7\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$ With this equation, you need to get rid of the $-2$ on the left-hand side so that the variable $x$ can be left on its own. If you add $2$ on the left-hand side, the constant term $-2$ will get cancelled out. Then, you also add $2$ to the right-hand side, to keep the equation in balance.

Rule

When adding or subtracting a term on both sides, you can look at it like you’re moving the term over to the other side and changing its sign. So I call this the “change side, change sign” rule.

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Activities 1