Solving Equation with Reduced Row Echelon Form

Question

I have the matrix in reduced row echelon form as:

$$\left[\begin{array}{ccc|c} 1 & 0 & 1 & 1/2 \\ 0 & 1 &-1 & -1/2 \\ 0 & 0 & 0 & 0 \\ \end{array}\right]$$

So, how do we find solutions based on this form?

Answer 1

My approach: Now we have $$x+z = \frac{1}{2}, \: y-z=-\frac{1}{2}$$ this can also be written as $$x = \frac{1}{2}-z, \: y = -\frac{1}{2}+z$$ and technically $z=z$ (this is obvious but just how I would think about it). Notice we only have two equations for three variables, so we will have infinitely many solutions. Our solutions are $\{(\frac{1}{2}-\lambda, -\frac{1}{2}+\lambda, \lambda): \lambda \in \mathbb{R}\}$