How do you solve the system x+y-z=2x+yz=2, x-2z=1x2z=1, and 2x-3y-z=82x3yz=8?

1 Answer
Dec 20, 2016

Write the coefficients of the equations into an augmented matrix and then perform Gauss-Jordan Elimination, using row operations.

Explanation:

Write the coefficients of the equations into an augmented matrix:

x - 2z = 1 to [(1, 0, -2,|,1)]

x + y - z = 2 to [(1, 0, -2,|,1),(1, 1, -1,|,2)]

2x - 3y - z = 8 to [(1, 0, -2,|,1),(1, 1, -1,|,2),(2, -3, -1,|,8)]

Perform Guass-Jordan Elimination using row operations on the augmented matrix:

[ (1, 0, -2,|,1), (1, 1, -1,|,2), (2, -3, -1,|,8) ]

Elementary Row Operations Notation

R_2 - R_1 to R_2

[ (1, 0, -2,|,1), (0, 1, 1,|,1), (2, -3, -1,|,8) ]

R_3 - 2R_1 to R_3

[ (1, 0, -2,|,1), (0, 1, 1,|,1), (0, -3, 3,|,6) ]

3R_2 + R_3 to R_3

[ (1, 0, -2,|,1), (0, 1, 1,|,1), (0, 0, 6,|,9) ]

R_3/6

[ (1, 0, -2,|,1), (0, 1, 1,|,1), (0, 0, 1,|,3/2) ]

R_2 - R_3 to R_2

[ (1, 0, -2,|,1), (0, 1, 0,|,-1/2), (0, 0, 1,|,3/2) ]

2R_3 + R_1 to R_1

[ (1, 0, 0,|,4), (0, 1, 0,|,-1/2), (0, 0, 1,|,3/2) ]

x = 4, y = -1/2, and z = 3/2

check:

4 - 2(3/2) = 1
4 + (-1/2) - 3/2 = 2
2(4) - 3(-1/2) - 3/2 = 8

1 = 1
2 = 2
8 = 8

This checks.