Step 1: Find the point of intersection of lines $x - 7y + 5 = 0$ and $3x + y = 0$. From the second equation, $y = -3x$. Substituting in the first equation: $x - 7(-3x) + 5 = 0$, which gives $x + 21x + 5 = 0$, so $22x = -5$ and $x = -\frac{5}{22}$.

Step 2: Find y-coordinate. Since $y = -3x$, we get $y = -3 \times (-\frac{5}{22}) = \frac{15}{22}$. The point of intersection is $(-\frac{5}{22}, \frac{15}{22})$.

Step 3: As per Example 4 in the context, the equation of a line parallel to y-axis passing through a point is $x =$ (x-coordinate of the point). Therefore, the required equation is $x = -\frac{5}{22}$ or $22x + 5 = 0$.