# How do you solve a simultaneous equation using matrices?

May 18, 2017

Write the equations as matrices.
Multiply both sides by the inverse matrix of the coefficients.

#### Explanation:

Here is the general principal, without using an actual example.

Equations such as $a x + b y = c \text{ "and" } \mathrm{dx} + e y = f$

can be written as matrices:

$\text{ } \left(\begin{matrix}a & b \\ d & e\end{matrix}\right) \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\begin{matrix}c \\ f\end{matrix}\right)$

A matrix multiplied by its inverse gives the unit or identity matrix
which is $\left(\begin{matrix}1 & 0 \\ 0 & 1\end{matrix}\right)$

Multiplying by this matrix has the same result as multiplying by $1$.

Let ${M}^{-} 1$ be the inverse of $\left(\begin{matrix}a & b \\ d & e\end{matrix}\right)$

Find this inverse matrix.

Then multiplying both sides by ${M}^{-} 1$ gives:

${M}^{-} 1 \left(\begin{matrix}a & b \\ d & e\end{matrix}\right) \left(\begin{matrix}x \\ y\end{matrix}\right) = {M}^{-} 1 \left(\begin{matrix}c \\ f\end{matrix}\right)$

$\left(\begin{matrix}x \\ y\end{matrix}\right) = {M}^{-} 1 \left(\begin{matrix}c \\ f\end{matrix}\right)$

When you multiply the right side you will have the values for $x \mathmr{and} y$ in the resulting matrix.