Slope and y intercept from equation

What I'd like to do in this video is a few more examples recognizing the slope and y-intercept given an equation.

So let's start with something that we might already recognize: let's say we have something of the form (y = 5x + 3).

What is the slope and the y-intercept in this example here? Well, we've already talked about that we can have something in slope-intercept form where it has the form (y = mx + b), where (m) is the slope (which people use the letter (m) for the slope) times (x) plus the y-intercept (which people use the letter (b) for).

So if we just look at this, (m) is going to be the coefficient on (x) right over there. So (m = 5); that is the slope, and (b) is just going to be this constant term, (+3). So (b = 3); so this is your y-intercept.

So that's pretty straightforward. But let's see a few slightly more involved examples.

Let's say if we had the form (y = 5 + 3x). What is the slope and the y-intercept in this situation? Well, it might have taken you a second or two to realize how this earlier equation is different than the one I just wrote here.

It's not (5x); it's just (5), and this isn't (3); it's (3x). So if you want to write it in the same form as we have up there, you can just swap the (5) and the (3x). It doesn't matter what or which one comes first; you're just adding the two.

So you could rewrite it as (y = 3x + 5), and then it becomes a little bit clearer that our slope is (3) (the coefficient on the (x) term), and our y-intercept is (5).

Y-intercept: let's do another example.

Let's say that we have the equation (y = 12 - x). Pause this video and see if you can determine the slope and the y-intercept.

All right, so something similar is going on here that we had over here. The standard form slope-intercept form we're used to seeing the (x) term before the constant term, so we might want to do that over here.

So we could rewrite this as (y = -x + 12) (or negative (x + 12)). From this, you might immediately recognize, okay, my constant term when it's in this form—that's my (b); that is my y-intercept. So that's my y-intercept right over there.

But what's my slope? Well, the slope is the coefficient on the (x) term, but all you see is a negative here. What's the coefficient? Well, you could view (-x) as the same thing as (-1x). So your slope here is going to be (-1).

Let's do another example. Let's say that we had the equation (y = 5x). What's the slope and y-intercept there?

At first, you might say, "Hey, this looks nothing like what we have up here." This is only—I only have one term on the right-hand side of the equality sign here; I have two.

But you could just view this as (5x + 0), and then it might jump out at you that our y-intercept is (0), and our slope is the coefficient on the (x) term; it is equal to (5).

Let's do one more example. Let's say we had (y = -7). What's the slope and y-intercept there?

Well, once again, you might say, "Hey, this doesn't look like what we had up here. How do we figure out the slope or the y-intercept?"

Well, we could do a similar idea. We could say, "Hey, this is the same thing as (y = 0 \cdot x - 7)," and so now it looks just like what we have over here.

You might recognize that our y-intercept is (-7) (y-intercept is equal to (-7)), and our slope is the coefficient on the (x) term; it is equal to (0).

And that makes sense: for a given change in (x), you would expect zero change in (y) because (y) is always (-7) in this situation.