Given that $a - 2b = c$, write $a$ as a column vector when $b = \begin{pmatrix}6\\11\end{pmatrix}$ a
From $a-2b=c$, add $2b$ to both sides to get $a=c+2b$. Since $2b=\begin{pmatrix}12\\22\end{pmatrix}$, we have $a=\begin{pmatrix}9\\-5\end{pmatrix}+\begin{pmatrix}12\\22\end{pmatrix}=\begin{pmatrix}21\\17\end{pmatrix}$.
What the equation is telling you
The vector equation $a-2b=c$ means that if you subtract twice the vector $b$ from $a$, you end up at $c$. To find $a$, we undo that subtraction.
Isolating $a$
Add $2b$ to both sides: $$a-2b+2b=c+2b \quad\Rightarrow\quad a=c+2b.$$
Computing $c+2b$
First double $b$: $$2b=2\begin{pmatrix}6\\11\end{pmatrix}=\begin{pmatrix}12\\22\end{pmatrix}.$$ Now add to $c$ component-wise: $$a=\begin{pmatrix}9\\-5\end{pmatrix}+\begin{pmatrix}12\\22\end{pmatrix}=\begin{pmatrix}9+12\\-5+22\end{pmatrix}=\begin{pmatrix}21\\17\end{pmatrix}.$$
Quick check
Verify $a-2b$: $$\begin{pmatrix}21\\17\end{pmatrix}-\begin{pmatrix}12\\22\end{pmatrix}=\begin{pmatrix}9\\-5\end{pmatrix}=c,$$ so the result is consistent.
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