System of three linear equations with unknown constant

Question

I have a linear system with three equations:

$$\begin{align} x + y + z &= 3a\\ x + 2y + (a+2)z &= a\\ x - (a+1)y - z &= 0 \end{align}$$

I want to find values for a where the system is inconsistent in addition to the values of $x, y, z$ and $a$ for where the system is consistent.

How should one approach a problem like this?

Thanks in advance.

Answer 1

With $$z=x-(a+1)y$$ we get $$2x+y-(a+1)y=3a$$ and $$2x-ay=3a$$ eliminating $x$ we obtain $$-\frac{3a(a+3)}{2}y=-\frac{3a(a+3)}{2}$$ Now you can do the rest!