# How do you solve Ax=B given A=((2, 3)) and B=(4)?

Jun 22, 2016

$x = \left(\begin{matrix}m \\ \frac{4 - 2 m}{3}\end{matrix}\right)$ for any value of $m$

#### Explanation:

If $x = \left(\begin{matrix}m \\ n\end{matrix}\right)$
then (by definition of matrix multiplication)
$\textcolor{w h i t e}{\text{XXX}} A x = \left(2 , 3\right) \times \left(\begin{matrix}m \\ n\end{matrix}\right) = \left(2 m + 3 n\right)$

and if $A x = B = \left(4\right)$

then
$\textcolor{w h i t e}{\text{XXX}} 2 m + 3 n = 4$

Simplifying
$\textcolor{w h i t e}{\text{XXX}} n = \frac{4 - 2 m}{3}$

Note that there are infinitely many solutions for $x$
a few obvious possibilities are:
$\textcolor{w h i t e}{\text{XXX}} x = \left(\begin{matrix}0 \\ \frac{4}{3}\end{matrix}\right) \mathmr{and} x = \left(\begin{matrix}2 \\ 0\end{matrix}\right) \mathmr{and} x = \left(\begin{matrix}5 \\ - 2\end{matrix}\right)$