Question

Solve the system of equations using matrices. Use the Gaussian elimination method with​ back-substitution.

x+y-z=-3
4x-y+z=8
-x+5y-4z=1

The solution set is ?

Answer 1

Explanation:

Ⅰ. x+y-z = -3

Ⅱ. 4x-y+z = 8

Ⅲ. -x+5y-4z = 1

Use equation Ⅰ to eliminate the x terms from equations Ⅱ and Ⅲ.

Ⅰ. x+y-z = -3

Ⅱ. -5y+5z = 20

Ⅲ. 6y-5z = -2

Divide equation Ⅱ by the coefficient of its y term:

Ⅰ. x+y-z = -3

Ⅱ. y-z = -4

Ⅲ. 6y-5z = -2

Use equation Ⅱ to eliminate the y terms from equation Ⅲ.

Ⅰ. x+y-z = -3

Ⅱ. y-z = -4

Ⅲ. z = 22

Divide equation Ⅲ by the coefficient of its z term:

Ⅰ. x+y-z = -3

Ⅱ. y-z = -4

Ⅲ. z = 22

Back-substitution

y = -4 - (-1)z = 18

x = -3 - (1)y - (-1)z = 1