Step 1: Use the direct coefficient-ratio test for parallel lines, instead of converting to slope-intercept form.
Two lines $a_1x + b_1y + c_1 = 0$ and $a_2x + b_2y + c_2 = 0$ are parallel exactly when:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \]
Write the given line in this form:
\[ 2x - 6y - 7 = 0 \]
so $a_1 = 2$, $b_1 = -6$, $c_1 = -7$.
Step 2: Test Option (C), $x - 3y = 7$, by writing it in the same form.
\[ x - 3y - 7 = 0 \]
so $a_2 = 1$, $b_2 = -3$, $c_2 = -7$.
Step 3: Compute all three ratios and compare.
\[ \frac{a_1}{a_2} = \frac{2}{1} = 2, \qquad \frac{b_1}{b_2} = \frac{-6}{-3} = 2, \qquad \frac{c_1}{c_2} = \frac{-7}{-7} = 1 \]
Here $\frac{a_1}{a_2} = \frac{b_1}{b_2} = 2$, but this is not equal to $\frac{c_1}{c_2} = 1$. This is exactly the condition for parallel lines.
Step 4: Briefly confirm why the other coefficient patterns fail this same test.
For $y = 3x - 7$ (that is, $3x - y - 7 = 0$), the ratio $\frac{a_1}{a_2} = \frac{2}{3}$ does not equal $\frac{b_1}{b_2} = \frac{-6}{-1} = 6$, so it fails. For $2x = 9 - 6y$ (that is, $2x + 6y - 9 = 0$), we get $\frac{b_1}{b_2} = \frac{-6}{6} = -1 \neq \frac{a_1}{a_2} = 1$, so it also fails. Only Option (C) satisfies the exact ratio condition for parallel lines.
Final Answer:
The line parallel to $2x - 6y = 7$ is $x - 3y = 7$, which matches Option (C).
\[ \boxed{x - 3y = 7} \]