# Simultaneous Equations

## Introduction

Solving simultaneous equations is a way of finding the value of two variables from two equations at the same time. To be good at this, you'll first need to be confident with solving linear equations.

## Explaining with an Example

Take $$x$$, $$y$$ and the equations $x=y+10 \\x=-y$ We have to ask ourselves "how do we find out what $$x$$ and $$y$$ are?" We need to use both equations to answer this! Firstly we must try to eliminate one of the variables. Notice that we have two expressions which are each equal to $$x$$, so we can set them equal to each other and have an equation only involving $$y$$. \begin{align} y+10&=x \\[6pt]x&=-y \\[6pt]y+10&=-y \end{align} Now we have an equation which we can solve to find $$y$$. \begin{align} y+10&=-y \\[6pt] 2y+10&=0 \\[6pt] 2y&=-10 \\[6pt] y&=-5 \end{align} Use this new information to find $$x$$; by substituting $$y=-5$$ into either of the original equations. \begin{align} x&=-y \\[6pt] y&=-5 \\[6pt] x&=-(-5)=5 \\[8pt] y=-5&\hspace{10pt}x=5 \end{align}

## Worked Example

Find $$x$$ and $$y$$ given that: \begin{alignat*}{2} 8x=&y+6\hspace{18pt}&&\text{(1)} \\[6pt] 5y-&8x=2&&\text{(2)} \\[15pt] 5y-2&=8x&&\text{[from equation (2)]} \\[6pt] 5y-2&=y+6&&\text{[combining (1) and (2)]} \\[6pt] 4y&=8 \\[6pt] y&=2 \\[12pt] 8x&=2+6&&\text{[from equation (1)]} \\[6pt] x&=1 \\[12pt] y=2\hspace{5pt}&\hspace{5pt}x=1 \end{alignat*}