# How do you solve using gaussian elimination or gauss-jordan elimination, 4x-3y+z=9, 3x+2y-2z=4, x-y+3z=5?

Feb 11, 2018

The values x=2, y=0, and z=1 satisfy the equations
$4 x - 3 y + z = 9 - - - - - - - - - \left(1\right)$
$3 x + 2 y - 2 z = 4 - - - - - - - - \left(2\right)$
$x - y + 3 z = 5 - - - - - - - - - - - \left(3\right)$

#### Explanation:

Gaussian elimination method:
$4 x - 3 y + z = 9 - - - - - - - - - \left(1\right)$
$3 x + 2 y - 2 z = 4 - - - - - - - - \left(2\right)$
$x - y + 3 z = 5 - - - - - - - - - - - \left(3\right)$

Eliminate x from (2)
Equation (2) becomes
$\left(2\right) - \frac{3}{4} \left(1\right)$
$3 - \frac{3}{4} \left(4\right) = 3 - 3 = 0$
$2 - \frac{3}{4} \left(- 3\right) = 2 + \frac{9}{4} = 2 + 2.25 = 4.25$
$- 2 - \frac{3}{4} \left(1\right) = - 2 - \frac{3}{4} = - 2 - 0.75 = - 2.75$
$4 - \frac{3}{4} \left(9\right) = 4 - 6.75 = - 2.75$
Changing the coefficients
$0 x + 4.25 y - 2.75 z = - 2.75$
$4.25 y - 2.75 z = - 2.75$

Eliminate x from (3)
Equation (3) becomes
$\left(3\right) - \frac{1}{4} \left(1\right)$
$1 - \frac{1}{4} \left(4\right) = 1 - 1 = 0$
$- 1 - \frac{1}{4} \left(- 3\right) = - 1 + 0.75 = - 0.25$
$3 - \frac{1}{4} \left(1\right) = 3 - 0.25 = 2.75$
$5 - \frac{1}{4} \left(9\right) = 5 - 2.25 = 2.75$
Changing the coefficients
$0 x - 0.25 y + 2.75 z = 2.75$
$- 0.25 y + 2.75 z = 2.75$
Thus the equations where x is eliminated are:
$4.25 y - 2.75 z = - 2.75 - - - - - - - - \left(4\right)$
$- 0.25 y + 2.75 z = 2.75 - - - - - - - - \left(5\right)$

Eliminate y from (5)
$\left(3\right) - \frac{- 0.25}{4.25} \left(4\right)$
$- 0.25 - \frac{- 0.25}{4.25} \left(4.25\right) = - 0.25 + 0.25 = 0$
$2.75 - \frac{- 0.25}{4.25} \left(- 2.75\right) = 2.75 + 0.162 = 2.912$
$2.75 - \frac{- 0.25}{4.25} \left(- 2.75\right) = 2.75 + 0.162 = 2.912$
Changing the coefficients
$0 y + 2.912 z = 2.912$
$2.912 z = 2.912$
Thus, the equation where y is eliminated is
$2.912 z = 2.912$-----------(6)
Solving for z,
$z = \frac{2.912}{2.912} = 1$

$z = 1$

Substituting the value z=1 in (4)
4.25y-2.75(1)=-2.7 Simplifying5
$4.25 y - 2.75 = - 2.75$
Transposing -2,75 from lhs to rhs
$4.25 y = - 2.75 + 2.75$
$4.25 y = 0$
Solving for y
$y = \frac{0}{4.25}$

$y = 0$

Substituting the values y=0, and z=1 in (1)
$4 x - 3 \left(0\right) y + \left(1\right) = 9$
Simplifying
$4 x - 0 + 1 = 9$
$4 x = 9 - 1$
$4 x = 8$
Solving for x

$x = \frac{8}{4}$

$x = 2$

Check:
$4 \left(2\right) - 3 \left(0\right) + \left(1\right) = 9 - - - - - - - - - \left(1\right)$
$3 \left(2\right) + 2 \left(0\right) - 2 \left(1\right) = 4 - - - - - - - - \left(2\right)$
$\left(2\right) - \left(0\right) + 3 \left(1\right) = 5 - - - - - - - - - - - \left(3\right)$
$8 - 0 + 1 = 9$, lhs=rhs
$6 + 0 - 2 = 4$, lhs=rhs
$2 - 0 + 3 = 5$, lhs=rhs

Thus, the values x=2, y=0, and z=1 satisfy the equations
$4 x - 3 y + z = 9 - - - - - - - - - \left(1\right)$
$3 x + 2 y - 2 z = 4 - - - - - - - - \left(2\right)$
$x - y + 3 z = 5 - - - - - - - - - - - \left(3\right)$