Simultaneous Equations


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