INSTRUCTOR: Welcome to friendly math 101. Today our lesson is on writing equations of perpendicular lines. In order for you to understand how to write equations of perpendicular lines, there are a few terms and phrases that we have to review. First, point-slope form-- point-slope form of an equation is y minus y1 equals m times x minus x1. And with this form, we will substitute values in for y1, m, and x1. 

Slope-intercept form is y equals mx plus b, where m is the slope and b is the y-intercept. Opposite reciprocal refers to two different things. When I say opposite during this tutorial, I mean opposite sign. So if it's positive, it becomes negative. Negative becomes positive. Reciprocal means we're going to flip our fraction. So 2/3 would become 3/2 or 3 over 2. 

Let's take a look at our first example. So if we have an equation y equals 2/3 x plus 1 and we have a point and we're asked to write an equation that's perpendicular to this equation but goes through the point 4, 3, first of all, we're looking at the equation, and we need to figure out what is perpendicular to that equation. We know that perpendicular lines have opposite reciprocal slopes. So the opposite of positive 2/3 would be a negative. And then the reciprocal would be 3/2. 

And that's all we need this equation for, so we could go ahead and cross that out. If we know that it goes through the point 4, 3, we're going to start off with point-slope form. So 4 is my x1. 3 is my y1. Substituting these values into point-slope form, we get y minus 3 equals negative 3/2 times x minus 4. If we're asked to write an equation in point-slope form, this would be our answer. But if we're asked to write it in slope-intercept form, we're going to have to solve for y. 

The first step is to distribute. So we're going to distribute negative 3/2. We get negative 3/2 x plus 6. And to get rid of the constant, we use inverse operations, and we get y equals negative 3/2 x plus 9. And this is our equation in slope-intercept form that goes through the point 4, 3 and is perpendicular to the equation y equals 2/3 x plus 1. 

Here's our second example. If we're given the equation y equals negative 4x plus 7 and we have to write an equation that goes through negative 2-- the opposite reciprocal of negative 4 is a little tricky because a lot of people don't understand that negative 4 is a fraction. We just have to write it over 1, and then it becomes a fraction. Any number over 1 doesn't change in value. 

So the opposite of negative 4 will be positive 1/4. And again, we're going to start with point-slope form, substituting these values in for our x1 and y1. x minus 3 here. When we see minus a negative, we're going to change that to plus. And if we're writing an equation in slope-intercept form, our first step is to distribute to get rid of the parentheses. So we get y plus 2 equals 1/4 x minus 3/4. 

And then we're going to solve for y. So we need to get rid of this positive 2 by applying an inverse operation. And our answer is y equals 1/4 x minus 11/4. And some of our answers might look a little bit messy because we're going to have fractions as our y-intercepts. And that's OK. 

Let's do one more example. So we're going to write an equation that's perpendicular to this equation and goes through the point negative 6, 8. The opposite reciprocal of negative 1/2, we're going to take the opposite sign. So it's going to be positive. And then to flip 1/2, that's going to become positive 2/1. Or the simplified version of that would just be 2. 

Again, starting with point-slope form, we get y minus 8 equals 2 times x minus a negative 6. If you need to review that, we do have some tutorials on point-slope form. Distribute the 2 to solve for y. We're converting this to slope-intercept form. And then we have to add 8 to get rid of our constant. And we get y equals 2x plus 20. So this is our equation that is perpendicular to our original equation but also goes through the point negative 6, 8. 

That concludes our lesson for today on writing equations of perpendicular lines. If you have any questions, please feel free to leave those questions in the comment section below. And for more math tutorials, subscribe to our channel, Friendly Math 101.