- So in this video, we're going to talk about, how do we determine if lines are parallel, perpendicular, or neither? So just in case you were wondering, this is a symbol for parallel lines. This is a symbol for perpendicular. So we're looking at these two equations that we're given. And we need to figure out, are they parallel, are they perpendicular, or are they a neither case? 

Now, the way that we do that is this. The test to determine if they're parallel or perpendicular has to do with the slope of the lines. You need to compare the slope. So if I'm looking at these equations, I need to figure out, how do I find the slope of these two lines? Because that's what I need to get to. The slope is the key here. That's what's going to let me know my answer. 

Now, because I'm given equations and I use slope-intercept form with almost every problem in dealing with graphing equations of lines, I'm going to use the slope-intercept form of a line here so I can get these equations put into slope-intercept form so it's a little easier to see what the slope is. 

So quick background just in case-- this is slope-intercept form of a line. To put it in slope-intercept form, you need to solve for y, get it completely solved by itself on one side. And then I'm just looking at m. m represents the slope of my line. So as long as I put the equation in this form, solve it for y, all I got to look is what's in front of my x. That's my slope. That's where I'm going to start. So I'm going to get both of these equations, solve for y. 

Over here on the left, all I got to do is subtract this 3x on both sides. 3x minus 3x cancels, leaving me with y is equal to negative 3x plus 4. So my slope here is a negative 3 because your slope is the coefficient of your x term, as long as it's in slope-intercept form. So slope is negative 3 for this equation. I'm going to pause there and work on the equation over here, get it solved for y. 

This is 3 times y. So here, we need to divide out everything by 3. 3 divided by 3-- we drop down the y-- is equal to-- you got to make sure you understand there's a 1 understood to be in front of that x. So this is technically 1/3 x minus-- 6 divided by 3 is a 2. I'm only concerned with my slope here. So my slope, again, is a coefficient of my x, here is a 1/3. 

So I have a slope of a negative 3, and I have a slope of 1/3. So the rule is, for parallel and perpendicular, say this. They're parallel if they have the same slope, same exact slope. So here, I have a negative 3. Here, I have a 1/3. Those are obviously not the same thing. They would both have to be negative 3 or both have to be 1/3. So negative 3 is not the same thing as 1/3. So since they don't perfectly match up, they are not parallel. 

Now, the rule for perpendicular says this. Take the slopes, multiply them together. If you get a negative 1, they are perpendicular. If you multiply these two slopes together, you get anything other than negative 1, they are not perpendicular. So what I need to do is I need to take negative 3, and I need to multiply it by 1/3. Now, since I'm dealing with a fraction here, I am going to do fraction multiplication. So I'm going to put that negative 3/1 so I can multiply across here. 

Negative 3 times 1 is a negative 3-- over 1 times 3 is a 3. If I simplify this, negative 3 divided by 3 is a negative 1. A negative 1 is the answer I'm looking for. Because I get a negative 1, that means that these two lines are perpendicular. Had I gotten anything else-- had I gotten a positive 1, had I gotten a 2, negative 10-- doesn't matter. Anything other than a negative 1 means they're not perpendicular. 

So had that happened-- we knew they weren't parallel already. Let's say we would have gotten a positive 1. That means they're not perpendicular. Then we would have just gone with the neither case, which is your most typical case. But we did get a negative 1, so we do know that our two lines are perpendicular together. That means that our lines cross, and they form a perfect 90-degree angle. Think about a four-way stop sign. Otherwise, that's it for this video.