Lesson Find the equation of a perpendicular line through a given point

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This lesson will go over an example problem detailing how to find the equation of a perpendicular line through a given point.

Problem:
Consider the equation 3x-7y = 19
Find the equation of the perpendicular line through (0.5, -2.5)

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Solution:
The given equation is of the form Ax+By = C
Anything perpendicular to this is of the form Bx-Ay = D
We have the A,B values swap places. Also, the new y coefficient becomes negated.

The given equation has A = 3 and B = -7, so
Bx-Ay = D
updates to
-7x-3y = D

Then we'll plug in the coordinates of (0.5, -2.5) to determine the value of D.
-7x-3y = D
-7(0.5)-3(-2.5) = D
-3.5+7.5 = D
4 = D
D = 4

Therefore, we go from
-7x-3y = D
to
-7x-3y = 4
as the final answer.

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Check:

To confirm that 3x-7y = 19 is perpendicular to -7x-3y = 4, find the slope of each
Recall that Ax+By = C has a slope of m = -A/B.
This means 3x-7y = 19 has a slope of m = -3/(-7) = 3/7
Meanwhile -7x-3y = 4 has a slope of m = -(-7)/(-3) = -7/3

Then note how the slopes multiply to -1
(3/7)*(-7/3) = -1
This property of multiplying to -1 works for any pair of perpendicular slopes (proof shown <a href="https://www.algebra.com/algebra/homework/word/misc/Miscellaneous_Word_Problems.faq.question.1207909.html">here</a>). This is as long as neither line is vertical nor horizontal.

Graph:
{{{drawing(500,500,-5,5,-5,5,
graph(500,500,-5,5,-5,5,-100,(19-3x)/(-7),(4+7x)/(-3)),
circle(0.5, -2.5, 0.08),
circle(0.5, -2.5, 0.1),
circle(0.5, -2.5, 0.12),
line(0.4, -2.2667,0.6398, -2.1639),
line(0.6398, -2.1639,0.7398, -2.3972),
locate(0.7, -2.55,"(0.5, -2.5)")
)}}}
3x-7y = 19 is in green
-7x-3y = 4 is in blue
These two lines form a 90 degree angle when meeting up.
You can use graphing tools like Desmos, GeoGebra, or anything similar to help confirm the answer.