This article was coauthored by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math instructor at the City College of San Francisco and was previously in the Math Department at Saint Louis University. She has taught math at the elementary, middle, high school, and college levels. She has an MA in Education, specializing in Administration and Supervision from Saint Louis University.
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Finding the equation of a line perpendicular to another line is a simple process that can be completed in two different ways. The first way is to solve for the equation of a line with one point and the equation of a line that runs perpendicular to it. The second way is to use two points from one line and one point from a perpendicular line. If a line runs perpendicular to another line, it means that it crosses it at a right angle and that you need to find its slope. The equation for a line on a graph is .[1] X Research source The is the line, the is the slope of the line multiplied by , and the is where the line intercepts the yaxis of the graph.[2] X Research source
Steps
Solving with One Point and an Equation

1Simplify the equation of the line. If you are given the equation of a line and one common point and asked to find a line that runs perpendicular to it, it is important that you first convert the equation into the format. To do this, you want to get the by itself.[3] X Research source
 For example, say your given equation is .
 To isolate , first move to the opposite side of the equation by adding it to both sides to get
 Get rid of the on the by dividing both sides of the equation by .
 The new equation will be .

2Calculate the opposite reciprocal of the slope. When a line is perpendicular to another line, the slope will be the negative opposite of the original line. This is called the opposite reciprocal. The lines cross each other at a right angle, so the slopes must be opposite. Two perpendicular slopes multiplied together will always equal .[4] X Research source
 Remember that represents the slope of the line.
 The opposite reciprocal of the equation would be or .
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3Plug the point into the slope equation to find the yintercept. Now that you have the slope of the perpendicular line, you can plug the value of the slope and the point you were given into a slope equation. This will give you the value of the yintercept. Using the yintercept, you can move on to complete the slope equation.[5] X Research source
 Remember that represents the yintercept of the line.
 For example, say your given point is where represents the coordinate and is the coordinate.
 Replace the letters in the equation with your known values of slope and xy coordinates:

4Solve the equation for the yintercept. Once you have your values entered into the slope equation, it is time to isolate , or the yintercept. To isolate , you must move all other numbers from one side of the equation. After you solve for the yintercept, you will know all of the numbers needed to write the equation of the perpendicular line.[6] X Research source
 To isolate in the equation , add to both sides.
 The equation for the yintercept of the perpendicular line will be

5Use the values for slope and yintercept to create your equation. Once you know the value for the slope and yintercept of your line, all you have to do is reassemble the numbers into the slope formula . Substitute the with the slope you have calculated and the with the yintercept you found.[7] X Research source
 The formula for the perpendicular line would come out to be
Calculating a Perpendicular Line Using Three Points

1Understand the coordinates you were given. If you are given three coordinates from two perpendicular lines, they cannot all be used for the same equations. The first two coordinates will be used for one line, and the third will be used once you begin calculating the equation of the perpendicular line. The goal is finding two perpendicular equations.[8] X Research source
 For example, you may be asked to find the coordinates of a line that passes through based on a line that passes through and .
 Focus on and for now.

2Substitute the points from the original line into the slope equation. You can use two separate points from one line to find the equation of a line that runs perpendicular to it. Before you calculate the equation of the perpendicular line, you will need to find the slope of the line that crosses the two points. The equation for finding the slope of a line with two points is . In this case, the numbers next to the and coordinates are not exponents, just a marker to show the different points.[9] X Research source
 If your points are and , then the slope would be
 simplifies to which is equal to .
 The slope of the line is

3Combine the two points with the slope into an equation. Once you know the value of the slope , you can use it to find the equation of your line by combining it with the and values. It does not matter which point you choose. The equation is . The exponents show the difference in the coordinates and do not represent any calculations.[10] X Research source
 Using the points , the equation would be .

4Simplify the equation to solve for . Once you have your chosen point and slope plugged into the equation, it is time to simplify. This will give you the equation of one line. After you know the equation of this line, you will be able to figure out the equation of the line that runs perpendicular to it.[11] X Research source
 To simplify , first multiply all of the numbers in parentheses by the outer value to get
 Isolate the on one side of the equation by adding to both sides to get . This is the equation of your first line.

5Find the slope of the perpendicular line using the opposite reciprocal. A line perpendicular to another line will always have an opposite slope. If the slope of the original line is a positive whole number, then the slope of the perpendicular line will be a negative fraction. Two perpendicular slopes multiplied together will always equal .[12] X Research source
 The opposite reciprocal of is .

6Solve the equation for the perpendicular line. Use the values for slope and the third point to find the equation of the perpendicular line. Now you know that the equation for the perpendicular line starts with , but you still do not know what the yintercept value is. By plugging your known point back into and adding the known value for , you will get your answer.[13] X Research source
 Using the coordinates from the perpendicular line fill in the equation: .
 Simplify the equation so that it reads
 Isolate the by adding to both sides.
 The equation now reads . This is the final equation for the perpendicular line.
Expert Q&A
References
 ↑ https://www.mathsisfun.com/equation_of_line.html
 ↑ https://www.mathopenref.com/perpendicular.html
 ↑ https://www.purplemath.com/modules/slopyint.htm
 ↑ https://www.khanacademy.org/math/geometry/hsgeoanalyticgeometry/hsgeoparallelperpendiculareq/v/findtheequationofaline
 ↑ https://www.khanacademy.org/math/geometry/hsgeoanalyticgeometry/hsgeoparallelperpendiculareq/v/findtheequationofaline
 ↑ https://www.varsitytutors.com/act_mathhelp/howtofindtheequationofaperpendicularline
 ↑ https://www.varsitytutors.com/act_mathhelp/howtofindtheequationofaperpendicularline
 ↑ https://www.varsitytutors.com/act_mathhelp/howtofindtheequationofaperpendicularline
 ↑ https://www.varsitytutors.com/act_mathhelp/howtofindtheequationofaperpendicularline