Question

Solve linear equations by Gaussian Elimination

2x-3y+z-w+u=0

4x-6y+2z-3w-u=-5

-2x+3y-2z+2w-u=3

Answer 1

Augmented matrix for given system of equations

Your matrix

	X1	X2	X3	X4	X5	b
1	2	-3	1	-1	1	0
2	4	-6	2	-3	-1	-5
3	-2	3	-2	2	-1	3

Make the pivot in the 1st column by dividing the 1st row by 2

	X1	X2	X3	X4	X5	b
1	1	-3/2	1/2	-1/2	1/2	0
2	4	-6	2	-3	-1	-5
3	-2	3	-2	2	-1	3

Eliminate the 1st column

	X1	X2	X3	X4	X5	b
1	1	-3/2	1/2	-1/2	1/2	0
2	0	0	0	-1	-3	-5
3	0	0	-1	1	0	3

Find the pivot in the 3rd column (inversing the sign in the whole row) and swap the 3rd and the 2nd rows

	X1	X2	X3	X4	X5	b
1	1	-3/2	1/2	-1/2	1/2	0
2	0	0	1	-1	0	-3
3	0	0	0	-1	-3	-5

Eliminate the 3rd column

	X1	X2	X3	X4	X5	b
1	1	-3/2	0	0	1/2	3/2
2	0	0	1	-1	0	-3
3	0	0	0	-1	-3	-5

Find the pivot in the 4th column in the 3rd row (inversing the sign in the whole row)

	X1	X2	X3	X4	X5	b
1	1	-3/2	0	0	1/2	3/2
2	0	0	1	-1	0	-3
3	0	0	0	1	3	5

Eliminate the 4th column

	X1	X2	X3	X4	X5	b
1	1	-3/2	0	0	1/2	3/2
2	0	0	1	0	3	2
3	0	0	0	1	3	5

Solution set:

x = 3/2 + (3/2)t - (1/2)s

y = t

z = 2 - 3s

w = 5 - 3s

u = s

s,t free parameter