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.
