# How do you solve this system of equations: 4x + y + 8z = 6 , 7x - y = 9z = - 10 , and - 6x + y - 8z = 8?

Oct 16, 2017

$x = - 5$
$y = 2$
$z = 3$

#### Explanation:

We're given 3 equations:

$\left(1\right) \text{ } 4 x + y + 8 z = 6$

$\left(2\right) \text{ } 7 x - y + 9 z = - 10$

$\left(3\right) \text{ } - 6 x + y - 8 z = 8$

Let's create two new equations:

$\left(1\right) + \left(2\right) = \left(4\right) \text{ "" } 11 x + 17 z = - 4$

$\left(2\right) + \left(3\right) = \left(5\right) \text{ "" } x + z = - 2$

Now, we can multiply equation $\left(5\right)$ by 11, and then subtract it from $\left(4\right)$. This will let us solve for $z$:

$\left(4\right) - 11 \times \left(5\right)$

$\left[11 x + 17 z = - 4\right] - \left[11 x + 11 z = - 22\right]$

$\left[6 z = 18\right]$

$z = 3$

Now that we know what $z$ is, we can plug it back into another equation (let's use equation $\left(5\right)$ for convenience) and solve for $x$:

$x + z = - 2$

$x + 3 = - 2$

$x = - 5$

Now that we know what $x$ is, we can plug $x$ and $z$ into another equation (let's use equation $\left(1\right)$) and solve for $y$:

$4 x + y + 8 z = 6$

$4 \left(- 5\right) + y + 8 \left(3\right) = 6$

$- 20 + y + 24 = 6$

$y = 2$

Therefore, the solution to our system of equations is:

$x = - 5$
$y = 2$
$z = 3$