# How do you solve the following system?: 16x +13y =7, -19x +15y = -2

Apr 25, 2018

$x = \frac{131}{7}$

$y = - \frac{2047}{91}$

#### Explanation:

1. $16 x + 13 y = 7$
2. $15 y - 19 x = - 2$

In equation 1, make $y$ the subject.

$13 y = 7 - 16 x \to y = \frac{7 - 16 x}{13}$

Inject that into equation 2.

$15 \left(\frac{7 - 16 x}{13}\right) - 19 x = - 2$

Solve for $x$.

$\frac{105 - 240 x}{13} - 19 x = - 2$

$\frac{105 - 240 x}{13} = 19 x - 2$

$105 - 240 x = 247 x - 26$

$247 x - 240 x = 105 + 26$

$7 x = 131$

$x = \frac{131}{7}$

Inject the value $x$ into the rearranged equation 1 to calculate the value of $y$.

$y = \frac{7 - 16 \left(\frac{131}{7}\right)}{13}$

$y = - \frac{2047}{91}$

Apr 25, 2018

$x = \frac{131}{487}$
$y = \frac{101}{487}$

#### Explanation:

$16 x + 13 y = 7$
$- 19 x + 15 y = - 2$

Multiply top equation by $15$ and second linear equation by $13$ to make the values of $y$ the same so they cancel out when subtracted from each other:

$240 x + 195 y = 105$
$- 247 x + 195 y = - 26$
which gives:
$487 x = 131$ when subtracted from each other.
$x = \frac{131}{487}$

Substitute $x$ into first original equation:

$16 \left(\frac{131}{487}\right) + 13 y = 7$
$\left(\frac{2096}{487}\right) + 13 y = 7$
$13 y = 7 - \left(\frac{2096}{487}\right)$
$13 y = \left(\frac{1313}{487}\right)$
$y = \frac{\left(\frac{1313}{487}\right)}{13}$
$y = \frac{101}{487}$