Originally Posted by

**SlipEternal** For line one, let's call the variable you use $\displaystyle t$ and the variable given in the answer $\displaystyle t'$. Then let's see when the two vectors are equal.

$\displaystyle \begin{pmatrix}2 \\ 2 \\ 4\end{pmatrix} + \begin{pmatrix}-\tfrac{1}{2} \\ \tfrac{1}{2} \\ 1\end{pmatrix}t = \begin{pmatrix}2 \\ 2 \\ 4\end{pmatrix} + \begin{pmatrix}-1 \\ 1 \\ 2\end{pmatrix}t'$ when $\displaystyle t=2t'$. Since the map $\displaystyle t \mapsto 2t$ is a bijection, you wind up with the same line either way.

Same thing for line 2.