Find the equation of the line passing through (-3, 5) and perpendicular to the line through the points (2, 5)(-3, 6).

Concept:

The equation of the line passing through (x1,y1) with the slope m is (y - y1) = m(x - x1).

The slope of line passing through the points (x1,y1) and (x2,y2) is = \(\rm \dfrac {y_2-y_1}{x_2-x_1}\)

The two non-vertical lines are perpendicular to each other only if and only if their slopes are negative reciprocals of each other.

Calculations:

The equation of the line passing through (x₁,y₁) with the slope m is (y - y₁) = m(x - x1).

The slope of line passing through the points (x1,y1) and (x2,y2) is = \(\rm \dfrac {y_2-y_1}{x_2-x_1}\)

The slope of line passing through the points (2, 5)(-3, 6) is = \(\rm \dfrac {6-5}{-3-2}\)

⇒ m₁ = \(\rm \dfrac {1}{-5}\) = \(\rm \dfrac {-1}{5}\)

The two non-vertical lines are perpendicular to each other only if and only if their slopes are negative reciprocals of each other.

The slope of perpendicular to the line through the points (2, 5)(-3, 6) is m₂ = \(\rm \dfrac {-1}{m_1}\)

⇒ m2 = 5 

The equation of the line passing through ( (x1,y1) = (-3, 5) and perpendicular to the line through the points (2, 5)(-3, 6) is 

(y - y1) = m(x - x1)

⇒(y - 5) = 5(x +3)

⇒5x - y + 20 = 0.