# How do you solve using gaussian elimination or gauss-jordan elimination, 4x_1 + 5x_2 + 2x_3 = 11, 2x_2 + 3x_3 - 4x_4 = -2, 2x_1 + x_2 + 3x_4 = 12, x_1 + x_3 + x_4 = 9?

Jan 18, 2016

${x}_{1} = 2$
${x}_{2} = - 1$
${x}_{3} = 4$
${x}_{4} = 3$

#### Explanation:

Initial Augmented Matrix:
[ ( 4.00, 5.00, 2.00, 0.00, 11.00), ( 0.00, 2.00, 3.00, -4.00, -2.00), ( 2.00, 1.00, 0.00, 3.00, 12.00), ( 1.00, 0.00, 1.00, 1.00, 9.00) ]

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 4.00 so pivot entry = 1
[ ( 1.00, 1.25, 0.50, 0.00, 2.75), ( 0.00, 2.00, 3.00, -4.00, -2.00), ( 2.00, 1.00, 0.00, 3.00, 12.00), ( 1.00, 0.00, 1.00, 1.00, 9.00) ]

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


[ ( 1.00, 1.25, 0.50, 0.00, 2.75), ( 0.00, 2.00, 3.00, -4.00, -2.00), ( 0.00, -1.50, -1.00, 3.00, 6.50), ( 0.00, -1.25, 0.50, 1.00, 6.25) ]

Pivot $\textcolor{b l a c k}{2}$
Pivot Row 2 reduced by dividing all entries by 2.00 so pivot entry = 1
[ ( 1.00, 1.25, 0.50, 0.00, 2.75), ( 0.00, 1.00, 1.50, -2.00, -1.00), ( 0.00, -1.50, -1.00, 3.00, 6.50), ( 0.00, -1.25, 0.50, 1.00, 6.25) ]

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


[ ( 1.00, 0.00, -1.38, 2.50, 4.00), ( 0.00, 1.00, 1.50, -2.00, -1.00), ( 0.00, 0.00, 1.25, 0.00, 5.00), ( 0.00, 0.00, 2.37, -1.50, 5.00) ]

Pivot $\textcolor{b l a c k}{3}$
Pivot Row 3 reduced by dividing all entries by 1.25 so pivot entry = 1
[ ( 1.00, 0.00, -1.38, 2.50, 4.00), ( 0.00, 1.00, 1.50, -2.00, -1.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 2.37, -1.50, 5.00) ]

$$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, 2.50, 9.50), ( 0.00, 1.00, 0.00, -2.00, -7.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 0.00, -1.50, -4.50) ]

Pivot $\textcolor{b l a c k}{4}$
Pivot Row 4 reduced by dividing all entries by -1.50 so pivot entry = 1
[ ( 1.00, 0.00, 0.00, 2.50, 9.50), ( 0.00, 1.00, 0.00, -2.00, -7.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 0.00, 1.00, 3.00) ]

$$Non-pivot rows reduced for pivot column
by subtracting appropriate multiple of pivot row 4 from each non-pivot row


[ ( 1.00, 0.00, 0.00, 0.00, 2.00), ( 0.00, 1.00, 0.00, 0.00, -1.00), ( 0.00, 0.00, 1.00, 0.00, 4.00), ( 0.00, 0.00, 0.00, 1.00, 3.00) ]