# How do you solve using gaussian elimination or gauss-jordan elimination, 3w-x=2y + z -4, 9x-y + z =10, 4w+3y-z=7, 12x + 17=2y-z+6?

Feb 18, 2016

$w = - \frac{183}{735}$

$x = - \frac{111}{49}$

$y = \frac{27297}{1470}$

$z = \frac{71967}{1470}$

#### Explanation:

Gaussian elimination involves the gradual elimination of variables from the set of equations until values can be found for each variable.
$3 w - x = 2 y + z - 4$
$9 x - y + z = 10$
$4 w + 3 y - z = 7$
$12 x + 17 = 2 y - z + 6$

First reorder the terms in all equations so that they all have the same order of variables.
$3 w - x - 2 y - z = - 4$
$9 x - y + z = 10$
$4 w + 3 y - z = 7$
$12 x - 2 y + z = - 11$

Adding the first two equations together gives
$3 w + 8 x - 3 y = 6$
Adding the second pair of equations gives
$4 w + 12 x + y = - 4$
Adding the middle pair of equations gives
$4 w + 9 x + 2 y = 17$

In a similar process, add three times the middle equation to the first equation
$15 w + 44 x = - 6$
Take twice the middle equation from the third equation
$- 4 w - 15 x = 9$

Now add four times the first equation to fifteen times the second equation
$176 x - 225 x = - 24 + 135$
$- 49 x = 111$

$x = - \frac{111}{49}$

Substitute this back into one of the final pair of equations to get $w$
$15 w = - 6 - \left(- \frac{111}{49}\right) = - \frac{183}{49}$
$w = - \frac{183}{735}$

$2 y = 17 - 4 \left(- \frac{183}{735}\right) - 9 \left(- \frac{111}{49}\right) = \frac{12495 - 183 + 14985}{735}$
$y = \frac{27297}{1470}$

$z = 10 - 9 x + y = 10 - 9 \left(- \frac{111}{49}\right) + \frac{27297}{1470}$
$= \frac{14700 + 29970 + 27297}{1470} = \frac{71967}{1470}$