## What is Solving an Equation?

Solving linear equations is the method of taking an equation such as \(5x+4=16-x\) and finding the
value of \(x\). The trick? Always do the same thing to each side of the equals sign at every stage.
The process involves **simplifying and rearranging**.

## Worked Example 1: Getting Started

\[\text{Find x:}\hspace{12pt}5x+4=16-x\] \[\begin{alignat*}{2} 5x+4&=16-x \\[6pt] 5x+4+x&=16-x+x\hspace{10pt}&&\text{[add x to both sides]} \\[6pt] 6x+4&=16&&\text{[simplify]} \\[6pt] 6x+4-4&=16-4&&\text{[subtract 4 from both sides]} \\[6pt] 6x&=12&&\text{[simplify]} \\[6pt] x&=\tfrac{12}{6}&&\text{[divide both sides by 6]} \\[6pt] x&=2&&\text{[simplify]} \end{alignat*}\] It's often useful to write down next to each line what you're doing like above, it helps you keep track and it a great use when you're checking your work to see if you slipped up somewhere!

## Worked Example 2: Do it Your Way

\[\text{Find x:}\hspace{12pt}5(x+2)=15x\]
**Method 1:**
\[\begin{alignat*}{2}
5(x+2)&=15x
\\[6pt] x+2&=3x\hspace{10pt}&&\text{[divide both sides by 5]}
\\[6pt] 2&=2x&&\text{[subtract x from both sides]}
\\[6pt] 1&=x&&\text{[divide both sides by 2]}
\end{alignat*}\]
**Method 2:**
\[\begin{alignat*}{2}
5(x+2)&=15x
\\[6pt] 5x+10&=15x\hspace{10pt}&&\text{[expand bracket]}
\\[6pt] 10&=10x&&\text{[subtract 5x from both sides]}
\\[6pt] 1&=x&&\text{[divide both sides by 10]}
\end{alignat*}\]
There's no right or wrong way to rearrange an equation. As you can see above, you can often
take more than one route and, assuming you don't make a mistake, you'll always get the same
answer! **Don't be afraid to play around, there's only so far wrong you can go**.

Now you've got all the tools you need to start solving simultaneous equations.