# How do you solve using gaussian elimination or gauss-jordan elimination, 3x - 3y + z = -5, -2x+7y= 15, 3x + 2y + z = 0?

Jun 1, 2016

$\left(x , y , z\right) = \left(- 4 , 1 , 10\right)$

#### Explanation:

Initial Augmented Matrix:
[ ( 3.00, -3.00, 1.00, -5.00), ( -2.00, 7.00, 0.00, 15.00), ( 3.00, 2.00, 1.00, 0.00) ]

Pivot Action $n$
pivot row = n; pivot column = n; pivot entry augmented matrix entry at (n,n)

$$1. convert pivot n row so pivot entry = 1
2. adjust non-pivot rows so entries in pivot column = 0


Pivot $\textcolor{b l a c k}{1}$
Pivot Row 1 reduced by dividing all entries by 3.00 so pivot entry = 1
[ ( 1.00, -1.00, 0.33, -1.67), ( -2.00, 7.00, 0.00, 15.00), ( 3.00, 2.00, 1.00, 0.00) ]

$$Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 1 from each non-pivot row


[ ( 1.00, -1.00, 0.33, -1.67), ( 0.00, 5.00, 0.67, 11.67), ( 0.00, 5.00, 0.00, 5.00) ]

Pivot $\textcolor{b l a c k}{2}$
Pivot Row 2 reduced by dividing all entries by 5.00 so pivot entry = 1
[ ( 1.00, -1.00, 0.33, -1.67), ( 0.00, 1.00, 0.13, 2.33), ( 0.00, 5.00, 0.00, 5.00) ]

$$Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 2 from each non-pivot row


[ ( 1.00, 0.00, 0.47, 0.67), ( 0.00, 1.00, 0.13, 2.33), ( 0.00, 0.00, -0.67, -6.67) ]

Pivot $\textcolor{b l a c k}{3}$
Pivot Row 3 reduced by dividing all entries by -0.67 so pivot entry = 1
[ ( 1.00, 0.00, 0.47, 0.67), ( 0.00, 1.00, 0.13, 2.33), ( 0.00, 0.00, 1.00, 10.00) ]

$$Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 3 from each non-pivot row


[ ( 1.00, 0.00, 0.00, -4.00), ( 0.00, 1.00, 0.00, 1.00), ( 0.00, 0.00, 1.00, 10.00) ]