How to find the value(s) of $a$ and $b$ such that the following system of equations... Find the value(s) of $a$ and $b$ such that the following system of equations,
$\begin{cases}x - 2y = a \\
3x - 6y = b\end{cases}$
(1) is inconsistent. (2) Which values of a and b make this system consistent and further, (3) what can you say about the number of solutions to the system.
I'm not sure if my process for part (1) of the question is correct.
I made an augmented matrix using the system of equations:
$$\begin{bmatrix}
1 & -2 &a \\ 3 & -6 &b
\end{bmatrix}$$
I manipulated the matrix two separate times to find when the system is inconsistent, for b --> R2 = R2 - 3R1,
$$\begin{bmatrix}
1 & -2 &a \\ 0 & 0 &b-3a
\end{bmatrix}$$
and a --> R1 = R1 - (1/3)R2
$$\begin{bmatrix}
0 & 0 &a - {b\over3} \\ 3 & -6 &b
\end{bmatrix}$$
So, the system is inconsistent when/if  $b = 3a$ and $a=b/3$?
Some general advice on how to approach the problem is greatly appreciated, thank you!

There is a way to verify if your answer is correct yourself - by substituting values back into the original problem!
Suppose $a=\frac{b}{3}$, for example $a=2$ and $b=6$. Now you have the equations
$$\begin{align*}
x-2y&=2\\
3x-6y&=6
\end{align*}$$
Are these consistent or inconsistent? Solve them and look at the definition of consistency.
