The following is a homework problem:
Let $\begin{align*} W_1 &= \{(a_1, a_2, a_3) \in\mathbb{R}^3 \mid a_1 = 3 a_2\text{ and }a_3 = -a_2\}\\ W_2 &= \{(a_1, a_2, a_3) \in \mathbb{R}^3 \mid a_1 - 4 a_2 - a_3 = 0\} \end{align*}$ Describe the intersection of W1 and W2 and observe that it is a subspace.
I realize that the intersection is $\{ (a_1, a_2, a_3) \in \mathbb{R}^3 \mid a_1 = 3 a_2\text{ and }a_3 = -a_2\text{ and }a_1 - 4 a_2 - a_3 = 0 \}.$ Further, I've observed that this seems to just be the set, $W_1$, based on the conditions. I just kind of haphazardly messed with the formulas to try to figure stuff out with no real methodology.
What is the method to solving these kinds of problems? How do I solve this and similar problems?