How do you solve the system -4x-5y-z=18, -2x-5y-2z=12, and -2x+5y+2z=4?

Mar 18, 2017

$\left\{x = - 4 , y = 0 , z = - 2\right\}$

Explanation:

$- 4 x - 5 y - z = 18 \text{ , } \left(1\right)$

$- 2 x - 5 y - 2 z = 12 \text{ , } \left(2\right)$

$- 2 x + 5 y + 2 z = 4 \text{ , } \left(3\right)$

$\text{let us sum the equations numbered as (2) and (3) so as to eliminate}$

$\text{the terms named 'y' and 'z'} .$

$- 2 x \cancel{- 5 y} \cancel{- 2 z} - 2 x \cancel{+ 5 y} \cancel{+ 2 z} = 12 + 4$

$\text{rearrange the equation above.}$

$- 2 x - 2 x = 12 + 4$

$- 4 x = 16 \text{ , } x = - 4$

$\text{let us write as x=4 in the equation numbered as (1)}$

$- 4 \left(- 4\right) - 5 y - z = 18$

$16 - 5 y - z = 18$

$- 5 y - z = 18 - 16$

$- 5 y - z = 2 \text{ , } \left(4\right)$

$\text{let us write as x=4 in the equation numbered as (2)}$

$- 2 \left(- 4\right) - 5 y - 2 z = 12$

$8 - 5 y - 2 z = 12$

$- 5 y - 2 z = 12 - 8$

$- 5 y - 2 z = 4 \text{ , } \left(5\right)$

$\text{now , subtract (5) from (4) }$

$- 5 y - z - \left(- 5 y - 2 z\right) = 2 - 4$

$\cancel{- 5 y} - z \cancel{+ 5 y} + 2 z = - 2$

$- z + 2 z = - 2$

z=-2"

$\text{now use (4)}$

$- 5 y - \left(- 2\right) = 2$

$- 5 y + 2 = 2$

$- 5 y = 0$

$y = 0$