Q.Find the distance between the two parallel lines $3x + 4y + 7 = 0$ and $3x + 4y - 8 = 0$.

The two lines already share identical $x,y$ coefficients, so apply $d = \dfrac{|c_1 - c_2|}{\sqrt{a^2+b^2}}$.

For two parallel lines written with identical coefficients of $x$ and $y$, $ax + by + c_1 = 0$ and $ax + by + c_2 = 0$, the distance between them is

Here both lines have $a = 3$ and $b = 4$ (they are genuinely parallel, as required), with $c_1 = 7$ and $c_2 = -8$:

$d = \dfrac{\lvert 7 - (-8) \rvert}{\sqrt{3^2 + 4^2}} = \dfrac{\lvert 15 \rvert}{\sqrt{9 + 16}} = \dfrac{15}{\sqrt{25}} = \dfrac{15}{5} = 3$ …

Computing $c_1 - c_2$ as $7 - 8 = -1$ by dropping the sign of $c_2 = -8$; correctly $7 - (-8) = 15$. Also, this shortcut only works after confirming the $x$ and $y$ coefficients are iden …