# How do you solve using gaussian elimination or gauss-jordan elimination, 6x+2y+7z=20, -4x+2y+3z=15, 7x-3y+z=25?

Jan 26, 2016

$x = \frac{325}{102}$

$y = - \frac{165495}{2601}$

$z = \frac{470}{51}$

#### Explanation:

$6 x + 2 y + 7 z = 20$
$- 4 x + 2 y + 3 z = 15$
$7 x - 3 y + z = 25$

Gaussian elimination is a process of solving simultaneous equations in more than two variables by repeatedly adding and/or subtracting the equations.
Observing that there is an element $+ 2 y$ in each of the first two equations, if we subtract the middle equation from the first one we get
$10 x + 4 z = 5$

If we add $3 \cdot$the second equation to $2 \cdot$the third equation

$- 12 x + 6 y + 9 z = 45$
$14 x - 6 y + 2 z = 50$

we get $2 x + 11 z = 95$

Using a similar process on these two equations in $x$ and $z$
$10 x + 4 z = 5$
$10 x + 55 z = 475$
$51 z = 470$

$z = \frac{470}{51}$

$x = \frac{5 - 4 z}{10} = \frac{255 - 1880}{510} = \frac{1625}{510} = \frac{325}{102}$

Select any one of the original equations to solve for $y$

$2 y = 20 - 6 \cdot \left(\frac{325}{102}\right) - 7 \cdot \left(\frac{470}{51}\right)$
$2 y = \frac{20 \cdot 102 \cdot 51 - 6 \cdot 325 \cdot 51 - 7 \cdot 102 \cdot 470}{102 \cdot 51}$

$y = \frac{104040 - 99450 - 335580}{5202}$
$y = - \frac{330990}{5202} = - \frac{165495}{2601}$