# How do you solve 4x + 5y - 2z = 23, -6x + 2y + 7z = -14, and 8x + 3y + 3z = 11?

May 1, 2018

$x = 1 , y = 3 , z = - 2$

#### Explanation:

$4 x + 5 y - 2 z = 23 \implies$ eq-1
$- 6 x + 2 y + 7 z = - 14 \implies$ eq-2
$8 x + 3 y + 3 z = 11 \implies$ eq-3

Eliminate $Z$ between 1 & 2: multiply by 7 & 2 respectively, then add the equations:

$28 x + 35 y - 14 z = 161$
$- 12 x + 4 y + 14 z = - 28$
$16 x + 39 y = 133 \implies$ eq-4

Eliminate $Z$ between 2 & 3: multiply by 3 & -7 respectively, then add the equations:

$- 18 x + 6 y + 21 z = - 42$
$- 56 x - 21 y - 21 z = - 77$
$- 74 x - 15 y = - 119 \implies$ eq-5

Eliminate $y$ between 4 & 5: multiply by 15 & 39 respectively, then add the equations, $y ' s$ cancel, solve for $x$:

$240 x + 585 y = 1995$
$- 2886 x - 585 y = - 4641$
$- 2646 x = - 2646$
$x = 1 \implies$ plug in eq-4 solve for $y$:
$39 y = 117$
$y = 3 \implies$ plug in for x & y in eq-1 solve for $z$:
$4 + 15 - 2 z = 23$
$- 2 z = 4$
$z = - 2$