Q.Find the equation of the line passing through $(1, -2)$ and perpendicular to the line $3x - 4y + 5 = 0$.

The perpendicular slope is the negative reciprocal of the given line's slope; then use point-slope form.

First read the slope of the given line $3x - 4y + 5 = 0$ using $m = -\dfrac{a}{b} = -\dfrac{3}{-4} = \dfrac{3}{4}$.

A line perpendicular to it has slope equal to the negative reciprocal:

Now apply the point-slope form through $(1, -2)$:

Multiply both sides by $3$ to clear the fraction:

$3(y + 2) = -4(x - 1) \ \Rightarrow\ 3y + 6 = -4x + 4 \ \Rightarrow\ 4x + 3y + 2 = 0$ …

A quicker route uses the general-form shortcut: a line perpendicular to $3x - 4y + 5 = 0$ can be written by swapping coefficients and changing a sign, as $4x + 3y + k = 0$. Substitutin …

Using the reciprocal without the negative sign ($\tfrac43$ instead of $-\tfrac43$), or misreading the given slope as $-\tfrac3 …