Question

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

x + y + z = -5
x - y + 3z = -1
4x + y + z = -2

Answer 1

In augmented matrix form, the system is equivalent to

`\begin{bmatrix}1&1&1&-5\\1&-1&3&-1\\4&1&1&-2\end{bmatrix}`

Subtract row 1 from row 2, and 4(row 1) from row 3:

`\begin{bmatrix}1&1&1&-5\\0&-2&2&4\\0&-3&-3&18\end{bmatrix}`

Divide through row 2 by -2, and through row 3 by -3:

`\begin{bmatrix}1&1&1&-5\\0&1&-1&-2\\0&1&1&-6\end{bmatrix}`

Subtract row 2 from row 3:

`\begin{bmatrix}1&1&1&-5\\0&1&-1&-2\\0&0&2&-4\end{bmatrix}`

Divide through row 3 by 2:

`\begin{bmatrix}1&1&1&-5\\0&1&-1&-2\\0&0&1&-2\end{bmatrix}`

The last row tells you
`\boxed{z=-2}`. Then

`y-z=-2\implies y+2=-2\implies\boxed{y=-4}`

and

`x+y+z=-5\implies x-4-2=-5\implies\boxed{x=1}`