# How do you solve 1/2x+y=5 and -x-2y=4 using matrices?

Feb 25, 2016

There are no solutions; the equations represent two parallel lines which do not intersect.

#### Explanation:

If you multiply the first equation by $\left(- 2\right)$ you will see that the new version of the first equation and the second equation have identical expressions on the left side but different values on the right side.

I you try to handle this as a matrix problem:

$\left(\begin{matrix}\frac{1}{2} & 1 & 5 \\ - 1 & - 2 & 4\end{matrix}\right)$

using Cramer's Rule (determinants)
$x = | {D}_{x} \frac{|}{|} D | \textcolor{w h i t e}{\text{XXX}} y = | {D}_{y} \frac{|}{|} D |$

$| D | = | \left(\frac{1}{2} , 1\right) , \left(- 1 , - 2\right) | = \frac{1}{2} \times \left(- 2\right) - \left(- 1\right) \times 2 = 0$

but division by $0$ is not valid, so $x$ and $y$ can not be evaluated.

using Gauss-Jorden Method
$\left(\begin{matrix}\frac{1}{2} & 1 & 5 \\ - 1 & - 2 & 4\end{matrix}\right)$

$\rightarrow \left(\begin{matrix}1 & 2 & 10 \\ - 1 & - 2 & - 4\end{matrix}\right)$

$\rightarrow \left(\begin{matrix}1 & 2 & 10 \\ 0 & 0 & 6\end{matrix}\right)$

both the $x$ and $y$ columns are zero, so the second row of this matrix implies:
$\textcolor{w h i t e}{\text{XXX}} 0 \cdot x + 0 \cdot y = 6$
which is clearly impossible.