#### Answer

$\textbf{r} = \langle 1,-3,4\rangle + t\langle4,-1,0 \rangle$

#### Work Step by Step

Equation of a Line: $\textbf{r} = \textbf{r}_0 + t\textbf{v}$
$\textbf{r}_0 = \langle 1,-3,4\rangle$
We can rewrite $\textbf{r}(t) = \langle 3+4t, 5-t,7\rangle$ to $\textbf{r}(t) = \langle 3,5,7\rangle + t\langle 4,-1,0\rangle$
We simply have to take the $\textbf{v}$ component of the given $\textbf{r}(t)$ to obtain a parallel line since the $\textbf{v}$ gives the direction of the line.
$\textbf{v} = \langle 4,-1,0\rangle$
$\textbf{r} = \langle 1,-3,4\rangle + t\langle4,-1,0 \rangle$