A line with slope $3$ intersects a line with slope $5$ at point  $(10,15)$. What is the distance between the $x$-intercepts of these two lines?
$\textbf{(A) } 2 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 7 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 20$
Using the point-slope form, the equation of each line is
\[y-15=3(x-10) \longrightarrow y=3x-15\]
\[y-15=5(x-10) \longrightarrow y=5x-35\]
Substitute in $y=0$ to find the $x$-intercepts.
\[0=3x-15\longrightarrow x=5\]
\[0=5x-35\longrightarrow x=7\]
The difference between them is $7-5=\boxed{\textbf{(A) \ } 2}$.