Question

#2. Solve the system of equations by any method. ( x + 2y + 3z + 4w = 5 J -5x - 4y + 3z + 2w = 1 1 x-y+z-w = 1 2x + y + 2z + w = 2 Answer: (x,y,z,w) =

Answer 1

Augmented matrix for the given system of equations

solution using Gauss-Jordan elimination

Your matrix

	X1	X2	X3	X4	b
1	1	2	3	4	5
2	-5	-4	3	2	1
3	1	-1	1	-1	1
4	2	1	2	1	2

Find the pivot in the 1st column in the 1st row

	X1	X2	X3	X4	b
1	1	2	3	4	5
2	-5	-4	3	2	1
3	1	-1	1	-1	1
4	2	1	2	1	2

Eliminate the 1st column

	X1	X2	X3	X4	b
1	1	2	3	4	5
2	0	6	18	22	26
3	0	-3	-2	-5	-4
4	0	-3	-4	-7	-8

Make the pivot in the 2nd column by dividing the 2nd row by 6

	X1	X2	X3	X4	b
1	1	2	3	4	5
2	0	1	3	11/3	13/3
3	0	-3	-2	-5	-4
4	0	-3	-4	-7	-8

Eliminate the 2nd column

	X1	X2	X3	X4	b
1	1	0	-3	-10/3	-11/3
2	0	1	3	11/3	13/3
3	0	0	7	6	9
4	0	0	5	4	5

Make the pivot in the 3rd column by dividing the 3rd row by 7

	X1	X2	X3	X4	b
1	1	0	-3	-10/3	-11/3
2	0	1	3	11/3	13/3
3	0	0	1	6/7	9/7
4	0	0	5	4	5

Eliminate the 3rd column

	X1	X2	X3	X4	b
1	1	0	0	-16/21	4/21
2	0	1	0	23/21	10/21
3	0	0	1	6/7	9/7
4	0	0	0	-2/7	-10/7

Make the pivot in the 4th column by dividing the 4th row by -2/7

	X1	X2	X3	X4	b
1	1	0	0	-16/21	4/21
2	0	1	0	23/21	10/21
3	0	0	1	6/7	9/7
4	0	0	0	1	5

Eliminate the 4th column

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

Solution set:

x = 4

y = -5

z = -3

w = 5

(x , y , z , w) = (4 , -5 , -3 , 5)