Question

Find the solutions of

2x₁-3x₂-7x₃+5x₄+2x₅=-2
x₁-2x₂-4x₅+3x₄+x₅=-2
2x₁+0x₂-4x₃+2x₄+x₃=3
x₁-5x₂-7x₃+6x₄+2x₅=-7

Answer 1

Augmented matrix for given system of equations

solution using Gauss-Jordan elimination

Your matrix

	X1	X2	X3	X4	X5	b
1	2	-3	-7	5	2	-2
2	1	-2	-4	3	1	-2
3	2	0	-4	2	1	3
4	1	-5	-7	6	2	-7

Find the pivot in the 1st column and swap the 2nd and the 1st rows

	X1	X2	X3	X4	X5	b
1	1	-2	-4	3	1	-2
2	2	-3	-7	5	2	-2
3	2	0	-4	2	1	3
4	1	-5	-7	6	2	-7

Eliminate the 1st column

	X1	X2	X3	X4	X5	b
1	1	-2	-4	3	1	-2
2	0	1	1	-1	0	2
3	0	4	4	-4	-1	7
4	0	-3	-3	3	1	-5

Find the pivot in the 2nd column in the 2nd row

	X1	X2	X3	X4	X5	b
1	1	-2	-4	3	1	-2
2	0	1	1	-1	0	2
3	0	4	4	-4	-1	7
4	0	-3	-3	3	1	-5

Eliminate the 2nd column

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

Find the pivot in the 5th column in the 3rd row (inversing the sign in the whole row)

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

Eliminate the 5th column

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

Solution set:

x1 = 1 + 2s - t

x2 = 2 - s + t

x3 = s

x4 = t

x5 = 1

s, t - free