Question

The graphs of $2y + x + 3 = 0$ and $3y + ax + 2 = 0$ are perpendicular. Solve for $a.$

Answer 1

Rewrite each line in slope-intercept form:
2y + x + 3 = 0 ⇒ y = -1/2 x - 3/2, so m1 = -1/2.
3y + ax + 2 = 0 ⇒ y = -a/3 x - 2/3, so m2 = -a/3.

Perpendicular lines satisfy m1·m2 = -1:
(-1/2)(-a/3) = -1 ⇒ a/6 = -1 ⇒ a = -6.