Slope from equation | Mathematics I | High School Math | Khan Academy

We've got the equation ( y + 2 = -2 \cdot x - 3 ), and what I want to do is figure out what is the slope of the line that this equation describes.

There's a couple of ways that you can approach it. What my brain wants to do is, well, I know a few forms where it's easy to pick out the slope. For example, if I can manipulate that equation to get it in the form ( y = mx + b ), well then I know that this ( m ) here, the coefficient on the ( x ) term, well that's going to be my slope, and ( b ) is going to be my ( y )-intercept. We cover that in many other videos.

Another option is to get into point-slope form. So the general framework or the general template for point-slope form is if I have an equation of the form ( y - y_1 = M \cdot (x - x_1) ), well then I immediately know that the line that this equation describes is going to have a slope of ( M ) once again.

Here, the ( y )-intercept doesn't jump out at you. Let me make sure you can read this. The ( y )-intercept doesn't jump out at you, but you know a point that is on this line in particular. You know that the point ( (x_1, y_1) ) is going to be on this line ( (x_1, y_1) ).

So let's look at our original example. It might immediately jump out at you that this is actually in point-slope form. You might say, "Well, okay, I see. I have a ( M ) in -x." ( x_1 ) would be 3, I have my slope here and that answers our question. Our slope would be -2, but here it says plus two. I have to subtract a ( y_1 ).

Well, you could just rewrite this so it says ( y - (-2) = -2 \cdot (x - 3) ). Then you see it's exactly this point-slope form right over here. So our slope right over there is -2. If I were to ask you, "Well, give me a point that sits on this line," you could say all right, ( x_1 ) would be 3 and ( y_1 ) would be -2. This point sits on the line. It's not the ( y )-intercept, but it's a point on the line. We know the slope is -2.

Now another way to approach this is to just manipulate it so that we get into slope-intercept form. So let's do that. Let's manipulate it so we get into slope-intercept form. The first thing my brain wants to do is distribute this -2. If I do that, I get ( y + 2 = -2x - 2 \cdot -3 + 6 ).

Then I can subtract two from both sides and then I get ( Y = -2x + 4 ). So here I am in slope-intercept form. Once again, I could say, "All right, my M here, the coefficient on the ( x ) term, is my slope." So my slope is -2.

Let's do another example. So here this equation doesn't immediately go into either one of these forms, so let's manipulate it. If it's in either one of them, I like to get into slope-intercept form. It's a little bit easier for my brain to understand. So let's do that.

Let us collect. Well, let's get the ( x )'s. Let's just isolate the ( y ) on the right-hand side since the ( 2y ) is already there. So let's add three to both sides. I'm just trying to get rid of this -3. So if we add three to both sides, on the left-hand side we have ( -4x + 10 = 2y ).

These cancel out—that was the whole point. Now, to solve for ( y ), we just have to divide both sides by two. So if we divide everything by two, we get ( -2x + 5 = Y ). So this is in slope-intercept form. I just have the ( y ) on the right-hand side instead of the left-hand side, but we have ( y = mx + b ).

So our ( m ) is the coefficient on the ( x ) term right over here, so our slope is once again -2. Here our ( y )-intercept is five, in case we wanted to know it.

Let's do one more example. One more example. All right, so once again this is in neither slope-intercept nor point-slope form to begin with, so let's just try to get it to slope-intercept form. And like always, pause the video and see if you can figure it out yourself.

All right, so let's get all the ( y )'s on the left-hand side isolated and all the ( x )'s on the right-hand side. So let me get rid of this -3x. I'm going to add 3x to both sides. Let's get rid of this 3y over here. Let's subtract 3y from both sides.

You could view this as I'm doing two steps at once, but once again, I'm trying to get rid of this -3x. So I add 3x to the left, but I have to do it to the right if I want to maintain the equality. If I want to get rid of this 3y, well, I subtract 3y from here, but I have to do it on the left-hand side if I want to maintain the equality.

So do I get that? Cancels out. ( 5y - 3y = 2y ) is equal to ( 2x + 3x = 5x ). Then these two characters cancel out.

So if I want to solve for ( y ), I just divide both sides by two and I get ( Y = \frac{5}{2}x ) and I'm done. You might say, "Wait, this doesn't look exactly like slope-intercept form! Where's my B?" Well, your ( B )—if you wanted to see it—you could just write ( +0 ).

( B ) is implicitly zero right over here. So your slope, your slope is going to be the coefficient on the ( x ) term. It's going to be ( \frac{5}{2} ). If you want to know your ( y )-intercept, well, it's zero. When ( x ) is zero, ( y ) is zero.