What is the equation of the line perpendicular to 2x+3y=1 and passing through the point (0,k)?

Question

The correct answer is 3x-2y=-2k, but I need an explanation of how to do this.

Answer 1

Since the line is perpendicular, its slope must be the negative reciprocal of the slope of the given line.

 

m = -1/(-2/3) = 3/2

 

y = mx + b

 

y = (3/2)x + b

 

It passes through (0,k), so

 

k = (3/2)(0) + b

b = k

 

y = (3/2)x + k

 

(3/2)x - y = - k

 

Multiply by 2 on both sides to put the equation in standard form.

 

3x - 2y = -2k

Answer 2

2x + 3y = 1

 

        3y = -2x + 1

 

          y = (-2/3)x + (1/3)  ← slope = -2/3

 

Perpendicular line has slope -1/ (-2/3) = 3/2

 

Equation of line with slope 3/2 and passing through (0,k):

 

     y - k = (3/2)(x - 0)

 

          y = (3/2)x + k

 

          So, 2y = 3x + 2k

 

                3x - 2y = -2k