Question

Can you solve this using matrices by row operations
4x+y-2z=-6
2x-3y+3z=9
x-2y=0

Answer 1

(x, y, z) = (0, 0, 3)

Explanation:

The augmented matrix for the system is ...

`\left[\begin{array}{cccc}4&1&-2&-6\\2&-3&3&9\\1&-2&0&0\end{array}\right]`

Subtract 4 times the 3rd row from the first row.

`\left[\begin{array}{cccc}0&9&-2&-6\\2&-3&3&9\\1&-2&0&0\end{array}\right]`

Subtract 2 times the 3rd row from the second row.

`\left[\begin{array}{cccc}0&9&-2&-6\\0&1&3&9\\1&-2&0&0\end{array}\right]`

Subtract 9 times the 2nd row from the first row.

`\left[\begin{array}{cccc}0&0&-29&-87\\0&1&3&9\\1&-2&0&0\end{array}\right]`

Now, the first row can be divided by -29 to give ...

`\left[\begin{array}{cccc}0&0&1&3\\0&1&3&9\\1&-2&0&0\end{array}\right]`

You can subtract 3 times this first row from the second row to get ...

`\left[\begin{array}{cccc}0&0&1&3\\0&1&0&0\\1&-2&0&0\end{array}\right]`

And add 2 times the second row to the third to get ...

`\left[\begin{array}{cccc}0&0&1&3\\0&1&0&0\\1&0&0&0\end{array}\right]`

This matrix now tells you (x, y, z) = (0, 0, 3).