Further Exercises: Linear systems

The linear system can be written down as a matrix and simplified by the Gauss-Jordan algorithm.

Since the last line contains no more coefficients, $0=2-2(a+4)$ or $a=-3$. The system of equations is thus "under-defined" and we can choose $x_3$ arbitrarily, e.g. $x_3=z$. We add this equation to the system as a new line and set $a=-3$.

That is, $a=-3$ and $x_1=z$, $x_2=1-z$, and $x_3=z$ with $z\in ℝ$.