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

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

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Answer 1 · 2015-10-08T02:16:28

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

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

$z = \frac{- 189}{57}$ $z=(-189)/57$

Explanation:

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

Solve equations (1) and (2)

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

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

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

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

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

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

Multiply (2) with $2$ $2$

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

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

Take equations (4) and (5)

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

Multiply equation (5) with $6$ $6$

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

$z = \frac{- 189}{57}$ $z=(-189)/57$

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

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

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

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

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

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

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

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

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