Perpendicular to 3x -2y = 3 and passing through (3,-7)

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Answer in standard form:  -2x-3y = 15

Answer in slope intercept form:   y = (-2/3)x - 5

This perpendicular line has slope -2/3 and y intercept -5

Explanation:

The given equation is in standard form Ax+By = C

We can see that A = 3, B = -2, and C = 3.

Rule:

If we are given an equation in the form Ax+By = C, then anything perpendicular to this is of the form Bx-Ay = D. Note the swap of A and B, and the sign change.

Furthermore, note how Ax+By = C solves to y = (-A/B)+C/B so it has slope -A/B. We can also see that Bx-Ay = D solves to y = (B/A)x - D/A and this has slope B/A.

The original line has slope -A/B and the perpendicular slope has slope B/A. The two slopes multiply to -1 assuming that A,B are nonzero.

With that rule set up, anything perpendicular to 3x-2y = 3 is of the form -2x-3y = D

To find the value of D, we plug in the coordinates of (x,y) = (3,-7) which is the point we want the perpendicular line to go through.

So,

-2x-3y = D

D = -2x-3y

D = -2(3)-3(-7) plug in x = 3 and y = -7

D = -6+21

D = 15

The equation of the perpendicular line in standard form is -2x-3y = 15

If you wanted to solve for y, and get the equation in slope intercept form, then follow the steps shown below.

-2x - 3y = 15

-3y = 2x+15

y = (2x+15)/(-3)

y = (2x)/(-3) + 15/(-3)

y = (-2/3)x - 5   which is slope intercept form

The slope of this perpendicular line is -2/3, while the slope of the original line is 3/2. The two slopes multiply to -1.