# How do you solve the following system?: x+3y-z=1, 7x+y+z=1, 4x-2y-z=11

Oct 28, 2017

$x = \frac{25}{26}$, $y = - \frac{37}{26}$ and $z = - \frac{56}{13}$

#### Explanation:

After summing first 2 equations for removing $z$

$x + 3 y - z + 7 z + y + z = 1 + 1$ or $8 x + 4 y = 2$

After summing second and third ones for removing $z$

$7 x + y + z + 4 x - 2 y - z = 1 + 11$ or $11 x - y = 12$

Now, I removed $y$ term from them. I also found $x$

$4 \cdot \left(11 x - y\right) + 8 x + 4 y = 4 \cdot 12 + 2$

$52 x = 50$, so $x = \frac{25}{26}$

Now, I found $y$ from first one.

$8 \cdot \frac{25}{26} + 4 y = 2$ or $4 y = - \frac{74}{13}$, hence $y = - \frac{37}{26}$

Now, I found $z$ from third one.

$4 \cdot \frac{25}{26} - 2 \cdot \left(- \frac{37}{26}\right) - z = 11$

$\frac{50}{13} + \frac{37}{13} - z = 11$

$z = \frac{87}{13} - 11 = - \frac{56}{13}$