<s>[INST] How do we express the equation corresponding to the intersection of the planes $x + y + z = 1$ and $x + 2y + 2z = 0$? If a system with three unknowns and two equations are such that
$$
\begin{align}
x+y+z=1&\\
x+2y+2z=0
\end{align}
$$
In the answer it says that this system can be represented as
$$
\begin{pmatrix}
x \\ y \\z
\end{pmatrix} = \begin{pmatrix}
2 \\-1\\0
\end{pmatrix}
+t\begin{pmatrix}
0\\-1\\1
\end{pmatrix}, t\in{\\R}
$$
But I am confused on how they came up to this parametric equation. Is there a general way of coming up with this parametric form?
 [/INST] The proposed system of equations corresponds to the intersection of two planes which are not parallel, also known as a line. In order to obtain one of its possible parametrizations, subtract the first equation from the second in order to obtain:
\begin{align*}
(x + 2y + 2z) - (x + y + z) = 0 - 1 & \Longleftrightarrow y + z = -1
\end{align*}
Hence we conclude that $x = 2$. Consequently, if we let $z = t\in\mathbb{R}$, it results that $y = -1 - t$.
Finally, one gets the desired equation of the intersecting line:
\begin{align*}
\begin{pmatrix}
x\\
y\\
z
\end{pmatrix} =
\begin{pmatrix}
2\\
-1 - t\\
t
\end{pmatrix} =
\begin{pmatrix}
2\\
-1\\
0
\end{pmatrix} + t
\begin{pmatrix}
0\\
-1\\
1
\end{pmatrix}
\end{align*}
Hopefully this helps!
</s>