If the lines y = 3x + 7 and 2y + px = 3 are perpendicular to each other, find the value of p.

Given lines,

⇒ y = 3x + 7 and 2y + px = 3

⇒ y = 3x + 7 and 2y = -px + 3

⇒ y = 3x + 7 and y = $-\dfrac{p}{2}x + \dfrac{3}{2}$

Comparing above equations with y = mx + c we get,

Slope of 1st line = 3

Slope of 2nd line = $-\dfrac{p}{2}$

Since,

Product of slopes of perpendicular lines = -1.

$\therefore 3 \times -\dfrac{p}{2} = -1 \\[1em] \Rightarrow -p = \dfrac{-1 \times 2}{3} \\[1em] \Rightarrow -p = \dfrac{-2}{3} \\[1em] \Rightarrow p = \dfrac{2}{3}.$

Hence, p = $\dfrac{2}{3}$.