Find the equation of a straight line perpendicular to the line 2x + 5y + 7 = 0 and with y-intercept -3.

Given equation of line,

⇒ 2x + 5y + 7 = 0

Converting it in the form y = mx + c,

⇒ 5y = -2x - 7

⇒ y = $-\dfrac{2}{5}x - \dfrac{7}{5}$.

Comparing with y = mx + c we get,

m = $-\dfrac{2}{5}$

Let slope of other line be m', since lines are perpendicular so,

⇒ m × m' = -1

$\Rightarrow -\dfrac{2}{5} \times m' = -1 \\[1em] \Rightarrow m' = \dfrac{5}{2}.$

Given, y-intercept = -3, putting values of slope and y-intercept in y = mx + c we get,

$\Rightarrow y = \dfrac{5}{2}x + (-3) \\[1em] \Rightarrow y = \dfrac{5x - 6}{2} \\[1em] \Rightarrow 2y = 5x - 6 \\[1em] \Rightarrow 5x - 2y - 6 = 0.$

Hence, the equation of the line is 5x - 2y - 6 = 0.