# How do you solve -2x - 4y + 2z = -8, -x - 5y +12z = 5, 3x + 5y - z = 10?

$x = - 9$ and $y = 8$ and $z = 3$

#### Explanation:

From the given equations
$- 2 x - 4 y + 2 z = - 8 \text{ " }$first equation
$- x - 5 y + 12 z = 5 \text{ " }$second equation
$3 x + 5 y - z = 10 \text{ " }$third equation

We eliminate the variable x first

multiply the terms of the second equation by 2 then subtract it from the first equation

$- 2 x - 4 y + 2 z = - 8 \text{ " }$first equation
$- 2 x - 10 y + 24 z = 10 \text{ " }$second equation
after subtraction
$6 y - 22 z = - 18$ fourth equation

multiply the original second equation by 3 then add to the third equation

$- 3 x - 15 y + 36 z = 15 \text{ " }$second equation
$3 x + 5 y - z = 10 \text{ " }$third equation

$- 10 y + 35 z = 25 \text{ " }$ fifth equation reducible to
$- 2 y + 7 z = 5 \leftarrow$reduced fifth equation
Multiply this fifth equation by 3 then add to the fourth equation

$- 6 y + 21 z = 15 \text{ " }$fifth equation
$6 y - 22 z = - 18$ fourth equation

the result is
$- z = - 3$
and
$z = 3$

Using the reduced fifth equation $- 2 y + 7 z = 5$ and $z = 3$ solve for y

$- 2 y + 7 z = 5$
$- 2 y + 7 \left(3\right) = 5$
$- 2 y + 21 = 5$
$- 2 y = - 16$
$y = 8$

Using the original second equation $- x - 5 y + 12 z = 5 \text{ }$and $y = 8$ and $z = 3$ solve for $x$

$- x - 5 y + 12 z = 5 \text{ }$
$- x - 5 \left(8\right) + 12 \left(3\right) = 5 \text{ }$
$- x - 40 + 36 = 5$
$- x = 9$
$x = - 9$

the solution is $x = - 9$ and $y = 8$ and $z = 3$

God bless...I hope the explanation is useful..