Question

How do you solve the system `2a+b=3, 5a=15, a+b+c=-1` using matrices?

Answer 1

Please see the explanation for the procedure.

Explanation:

Write `5a = 15` into the first row of the augmented matrix:

`[ (5, 0,0,|,15) ]`

Add the row for `2a + b = 3`:

`[ (5, 0,0,|,15), (2, 1, 0,|,3) ]`

Add the row for `a + b + c = -1`:

`[ (5, 0,0,|,15), (2, 1, 0,|,3), (1,1,1,|,-1) ]`

Divide the first row by 5:

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

Multiply the first row by -2 and then add to row 2:

`[ (1, 0,0,|,3), (0, 1, 0,|,-3), (1,1,1,|,-1) ]`

Subtract row 1 from the row 3:

`[ (1, 0,0,|,3), (0, 1, 0,|,-3), (0,1,1,|,-4) ]`

Subtract row 2 from row 3:

`[ (1, 0,0,|,3), (0, 1, 0,|,-3), (0,0,1,|,-1) ]`

`a = 3, b = -3, c = -1`

Check:

`2(3) + -3 = 3`
`5(3) = 15`
`3 + -3 + -1 = -1`

`3 = 3`
`15 = 15`
`-1 = -1`

This checks