The distance between the line 3x + 4y + 5 = 0 and 3x + 4y - 7 = 0 is
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Before using the parallel distance formula, ensure the $A$ and $B$ coefficients are identical in both equations. If one is a multiple of the other, divide or multiply to make them match.
Step 1: Both lines have the same coefficients of \(x\) and \(y\) (3 and 4), confirming they are parallel, so a single unit normal vector, \(\left(\dfrac{3}{5}, \dfrac{4}{5}\right)\), points perpendicular to both.
Step 2: Write both lines in the form \(3x+4y = k\): the first gives \(k_1 = -5\) and the second gives \(k_2 = 7\).
Step 3: The perpendicular distance between the lines equals the difference in these \(k\) values divided by the magnitude of the normal direction, \(\sqrt{3^2+4^2} = 5\).
\[ d = \frac{|k_2 - k_1|}{5} = \frac{|7-(-5)|}{5} = \frac{12}{5} = \boxed{\dfrac{12}{5}} \]