How do you solve 3u + v + w = 93u+v+w=9, u + v - w = 5u+v−w=5 and u + 2v + w = 9u+2v+w=9 using matrices?

1 Answer
Eddie
Mar 5, 2017

((u),(v),(w)) = ((7/4),(7/2),(1/4))

Explanation:

In matrix form:

((3,1,1),(1,1,-1),(1,2,1)) ((u),(v),(w)) = ((9),(5),(9))

Using Gaussian elimination, we can row reduce, so we set up the augmented matrix:

((3,1,1),(1,1,-1),(1,2,1)) ((9),(5),(9))

R2 to R2 - R3

((3,1,1),(0,-1,-2),(1,2,1)) ((9),(-4),(9))

R3 to R3 -1/3 R1

((3,1,1),(0,-1,-2),(0,5/3,2/3)) ((9),(-4),(6))

R3 to R3 + 5/3 R2

((3,1,1),(color(red)(0),-1,-2),(color(red)(0),color(red)(0),-8/3)) ((9),(-4),(-2/3))

This is now upper triangular, we are in row echelon form. Now we back-substitute:

From the bottom row:

-8/3 w = -2/3

implies w = 1/4

From the middle row:

- v - 2 w =- 4

implies v = 4 - 2w

implies v = 7/2

From the top row:

3u + v + w = 9

implies u = (9 - v - w)/3

= (9 - 7/2 - 1/4)/(3)

implies u = 7/4

So:

((u),(v),(w)) = ((7/4),(7/2),(1/4))

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