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As the lines $L_{1}$ and $L_{2}$ are parallel, the line $L_{1}$, must have the same slope as $L_{2}.$ Therefore, we can start the equation by substituting $m$ with $4$ in the slope-intercept form.
$y=4x+b$
To find the $y$-intercept, $b,$ we'll use the given point $(4,17)$. Substitute this point's coordinates into the equation $y=4x+b$, and then solve for $b.$
The line $L_{1}$ has the equation $y=4x+1.$

$y=4x+b$

$17=4⋅4+b$

Solve for $b$

$b=1$