# How do solve the following linear system?:  -10x+7y=-2 , -4x-15y=-1 ?

Mar 5, 2016

$x = \frac{37}{178}$
$y = \frac{1}{89}$

#### Explanation:

Scale either equation (or both) with the goal of eliminating one of the variables

$\left[1\right] - 10 x + 7 y = - 2$

$\left[2\right] - 4 x - 15 y = - 1$

Multiply $\left[1\right]$ by -2 and multiply $\left[2\right]$ by 5

$\left[1 '\right] - 2 \left(- 10 x + 7 y = - 2\right)$

$\implies \left[1 '\right] 20 x - 14 y = 4$

$\left[2 '\right] 5 \left(- 4 x - 15 y = - 1\right)$

$\implies \left[2 '\right] - 20 x - 75 y = - 5$

$\left[1 '\right] 20 x - 14 y = 4$
$\left[2 '\right] - 20 x - 75 y = - 5$

If we add both equations, $x$ will be eliminated and we can solve for y

$\left[3\right] - 89 y = - 1$

$\implies y = \frac{1}{89}$

To get $x$, substitute the obtained value for $y$ in one of the equations $\left[1\right]$, $\left[1 '\right]$, $\left[2\right]$, $\left[2 '\right]$. For example, let's use $\left[1\right]$

$- 10 x + 7 y = - 2$

$\implies - 10 x + 7 \left(\frac{1}{89}\right) = - 2$

$= : > - 10 x + \frac{7}{89} = - 2$

$\implies 10 x = 2 + \frac{7}{89}$

$\implies 10 x = \frac{178 + 7}{89}$

$\implies 10 x = \frac{185}{89}$

$\implies x = \frac{37}{178}$