# How do you solve the following linear system -x + 2y = 6 , x - 4y = -3 ?

Dec 14, 2015

$\textcolor{w h i t e}{\times} x = - 9 \textcolor{w h i t e}{x} \text{and} \textcolor{w h i t e}{x} y = - \frac{3}{2}$

#### Explanation:

$\textcolor{w h i t e}{\times} - x + 2 y = 6 , x - 4 y = - 3$

With a different approach:
$\textcolor{w h i t e}{\times} - x + 2 y = 6 \iff \textcolor{red}{y = \frac{x + 6}{2}}$
and
$\textcolor{w h i t e}{\times} x - 4 y = - 3 \iff \textcolor{red}{y = \frac{x + 3}{4}}$

$\textcolor{w h i t e}{\times} \frac{x + 6}{2} = y$
$\implies \frac{x + 6}{2} = \textcolor{b l u e}{\frac{x + 3}{4}}$

Multiply both side by $\textcolor{g r e e n}{4}$ to simplify right side:
$\textcolor{w h i t e}{\times} \textcolor{g r e e n}{4 \times} \frac{x + 6}{2} = \textcolor{g r e e n}{4 \times} \frac{x + 3}{4}$
$\implies \textcolor{red}{2} x + \textcolor{red}{12} = x + 3$

Finally add $\textcolor{red}{- x - 12}$ to both side to find $x$.
$\implies 2 x + 12 \textcolor{red}{- x - 12} = x + 3 \textcolor{red}{- x - 12}$
$\implies x = - 9$

$\textcolor{w h i t e}{\times} y = \frac{x + 6}{2}$
$\textcolor{w h i t e}{\times x} = \frac{\textcolor{b l u e}{- 9} + 6}{2}$
$\textcolor{w h i t e}{\times x} = - \frac{3}{2}$