# Solving equations

## In a nutshell

An equation can be solved by rearranging. You usually want to find the value of the unknown variable, e.g $x$, which makes the equation true. Think of an equation as a balancing scale, whatever you do to one side of the equation, you must do the same to the other.

## Balancing scales

The equation

$x+3=7$

can be represented by:

To solve for $x$, you need to rearrange the equation so that $x$ is by itself on one side of the equation. The other terms and numbers need to be moved onto the other side of the equation. In this case it is necessary to move the $+3$ to the other side of the equation. To move $+3$ to the other side, do the opposite, or inverse function, to both sides of the equation. The inverse of $+3$ is $-3$. So subtract $3$ from both sides.

Consider the balancing scale, if you take 3 from the left hand side, you also need to take 3 from the right hand side. This leaves

$x=7-3 \\ \underline {x=4}$

which can be represented by:

## Solve algebraically

When solving an equation, think about what to do to both sides of the equation to get the unknown variable by itself.

$\begin{aligned} \quad x+3&=7 \\ -3 \qquad & \qquad -3 \end{aligned}\\\quad \quad \underline{x=4}$

When there is more than one number to move to the other side of the equation, it is important to move them in the right order. Use reverse BIDMAS to help.

##### Example 1

*Solve the equation*

$3x+4=16$

*Here, the *$3$* with the *$x$* and *$+4$* need to be moved to the other side. To decide which number to move across first, think about substituting a number in for *$x$*. You would take the value of *$x$*, multiply by *$3$* first, then *$+4$* to the result. So when solving an equation, reverse the process. So move *$+4$* to the other side first, then move *$3$* to the other side.*

*The opposite of *$+4$* is *$-4$*, so subtract *$4$* from both sides. Then *$3$* is multiplying *$x$*, so do the inverse and divide by *$3$* to both sides.*

$\begin{aligned}\quad 3x+4 &=16 \\-4 \qquad & \qquad -4 \\3x &=12 \\\div 3 \qquad & \qquad \div 3 \end{aligned} \\ \qquad \quad \underline{x=4}$

## Inverse operations

Addition ($+$) is the inverse of subtraction ($-$), multiplication ($\times$) is the inverse of division ($\div$) and squaring ($x^2$) is the inverse of square rooting ($\sqrt x$).