Subspace formed by coefficients of a linear equation

Question

Suppose I have the linear equation \begin{align} ax + by = c \hspace{10mm}(1) \end{align}

where $(a,b,c) \in \mathbb{R}^3$. Let $W = \{ (a,b,c) \in \mathbb{R^3} : (1)$ is consistent $\}$.

Is $W$ a subspace of $\mathbb{R^3}$ ?

Apparently the answer is no, but I am unable to find a counter-example. If anything, my reasoning (below) tells me $W$ is a subspace.

1. $(0,0,0) \in W$.
2. If $(a,b,c) \in W$, then $(\alpha a,\alpha b,\alpha c) \in W$ for any non-zero $\alpha \in \mathbb{R}$.
3. If $(a,b,c),(a',b',c') \in W$, then $(a + a', b + b', c+c') \in W$

Answer 1

$(1,-1,0)$ and $(-1,0,1)$ are consistent, but $(1,-1,0)+(-1,0,1) = (0,0,1)$ is inconsistent.