Step 1: Find the slope of the given line $x - 7y + 5 = 0$. Rearranging into slope-intercept form ($y = mx + c$): $7y = x + 5 \Rightarrow y = \frac{1}{7}x + \frac{5}{7}$. The slope is $\frac{1}{7}$.

Step 2: Determine the slope of the perpendicular line. The slope $m$ of the line perpendicular to the given line is the negative reciprocal: $m = -7$.

Step 3: Use the intercept form. The line has an $x$-intercept of 3, so $a = 3$. The equation is of the form $\frac{x}{a} + \frac{y}{b} = 1$. Substituting $a=3$, we get $\frac{x}{3} + \frac{y}{b} = 1$.

Step 4: Find the y-intercept ($b$). The slope of the line $\frac{x}{3} + \frac{y}{b} = 1$ can be written as $y = -\frac{b}{3}x + b$. The slope is $-\frac{b}{3}$. Equating this to the perpendicular slope found in Step 2: $-\frac{b}{3} = -7 \Rightarrow b = 21$.

Step 5: Write the final equation. Substituting $a=3$ and $b=21$ into the intercept form: $\frac{x}{3} + \frac{y}{21} = 1$. Multiplying by 21 gives $7x + y = 21$ or $7x + y - 21 = 0$.