[Math] Can’t solve 3 variables – Systems of Linear Equations

Question

I've was asked to solve this (as homework):

$$2x + y + z = 3$$
$$4x + 2z = 10$$
$$2x + 2y = -2$$

I need to solve it with matrices and I have NO IDEA how to do so.

I need your help. thanks.

Answer 1

Your system of equations is equivalent to: $$ \left( \begin{array}{ccc} 2 & 1 & 1 \\ 4 & 0 & 2 \\ 2 & 2 & 0 \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} 3 \\ 10 \\ -2 \end{array} \right). $$ You want to add/subtract multiples of the three rows (see Gaussian elimination) until you get to $$ \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} x \\ y \\ z \end{array} \right) = \left( \begin{array}{c} a \\ b \\ c \end{array} \right). $$