# How do you solve x+3y-z=13, 2x-5z=23, 4x-y-2z=14?

Oct 8, 2015

$x = \frac{366}{114}$

$y = \frac{738}{342}$

$z = \frac{- 189}{57}$

#### Explanation:

$x + 3 y - z = 13$ ------------------(1)
$2 x - 5 z = 23$------------------(2)
$4 x - y - 2 z = 14$ ----------------(3)

Solve equations (1) and (2)

$x + 3 y - z = 13$ ------------------(1)
$2 x - 5 z = 23$------------------(2)

Multiply equation (1) with $2$ to eliminate $x$ term

$x + 3 y - z = 13 \times 2$ ------------------(1)
$2 x + 0 y - 5 z = 23$------------------(2)

$2 x + 6 y - 2 z = 26$ ------------------(1) Subtract (2) from (1)
$2 x + 0 y - 5 z = 23$------------------(2)
$6 y + 3 z = 3$ ----------------(4)

Solve equations (2) and (3) and eliminate $x$ term

$2 x + 0 y - 5 z = 23$------------------(2)
$4 x - y - 2 z = 14$ ----------------(3)

Multiply (2) with $2$

$2 x + 0 y - 5 z = 23 \times 2$------------------(2)
$4 x - y - 2 z = 14$ ----------------(3)

$4 x + 0 y - 10 z = 46$------------------(2) Subtract (3) from (2)
$4 x - y - 2 z = 14$ ----------------(3)
$y - 8 z = 32$ ----------------(5)

Take equations (4) and (5)

$6 y + 3 z = 3$ ----------------(4)
$y - 8 z = 32$ ----------------(5)

Multiply equation (5) with $6$

$6 y + 3 z = 3$ ----------------(4) Subtract (5) from (4)
$6 y - 54 z = 192$ ----------------(5)
$57 z = - 189$

$z = \frac{- 189}{57}$

Substitute the value of $z$ in equation (2)

$2 x - 5 z = 23$------------------(2)
$2 x - 5 \left(\frac{- 189}{57}\right) = 23$------------------(2)
$2 x + \frac{945}{57} = 23$------------------(2) Multiply both sides by $57$
$114 x + 945 = 1311$-------- Solve it for $x$
$114 x = 1311 - 945 = 366$

$x = \frac{366}{114}$

Substitute the value of $x$ and $z$ in equation (1)

$x + 3 y - z = 13$ ------------------(1)
$\frac{366}{114} + 3 y - \frac{- 189}{57} = 13$
$\frac{366}{114} + 3 y + \frac{189}{57} = 13$ ---- Multiply both sides with $114$
$366 + 342 y + 378 = 1482$-------solve it for $y$

$744 + 342 y = 1482$
$342 y = 1482 - 744 = 738$

$y = \frac{738}{342}$