8th grade (Illustrative Mathematics)
Sal graphs the equation of a line that represents a proportional relationship given a unit rate. Created by Sal Khan.
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Graph the line that represents a proportional relationship between y and x with a unit rate 0.4. That is, a change of one unit in x corresponds to a change of 0.4 units in y. And they also ask us to figure out what the equation of this line actually is. So let me get my scratch pad out and we could think about it. So let's just think about some potential x and y values here. So let's think about some potential x and y values. So when we're thinking about proportional relationships, that means that y is going to be equal to some constant times x. So if we have a proportional relationship, if you have zero x's, it doesn't matter what your constant here is, you're going to have zero y's. So the point 0, 0 should be on your line. So if this is the point 0, 0, this should be on my line right over there. Now, let's think about what happens as we increase x. So if x goes from 0 to 1, we already know that a change of 1 unit in x corresponds to a change of 0.4 units in y. So if x increases by 1, then y is going to increase by 0.4. It's not so easy to graph this 1 comma 0.4. The 0.4 is hard to graph on this little tool right over here. So let's try to get this to be a whole number. So then when x increases another 1, y is going to increase by 0.4 again. It's going to get to 0.8. When x increases again by 1, then y is going to increase by 0.4 again. It's going to get to 1.2. If x increases again, y is going to increase by 0.4 again. So just to 1.6. Notice, every time x is increasing by 1, y is increasing by 0.4. That's exactly what they told us here. Now, if x increases by 1 again to 5, then y is going to increase 0.4 to 2. And I like this point because this is nice and easy to graph. So we see that the point 0, 0 and the point 5 comma 2 should be on this graph. And I could draw it. And I'm going to do it on the tool in a second as well. So it'll look something like this. And notice the slope of this actual graph over here. Notice the slope of this actual graph. If our change in x is 5. So notice, here our change in x is 5. Our change in x is 5. You see that as well. When you go from 0 to 5, this change in x is 5. Change in x is equal to 5. What was our corresponding change in y? Well, our corresponding change in y when our change in x was 5, our change in y was equal to 2. And you see that here, when x went from 0 to 5, y went from 0 to 2. So our change in y in this circumstance is equal to 2. So our slope, which is change in y over change in x, is the rate of change of your vertical axis with respect to your horizontal axis, is going to be equal to 2 over 5, or 2/5. Which if you wrote it as a decimal is equal to 0.4. So this right over here is your slope. So I'm going to do this with the tool. But first, let's also think about what the equation of this line is going to be. Well, we know that y is equal to some constant times x. And we know that the point 5, 2 is on this line right over here. So we could say, well, when x is equal to 5, y is equal to 2. Or, when y equals 2, we have k times 5, or k is equal to-- dividing both sides by 5, you can't see that. If I divide both sides by 5, I'm left with k is equal to 2/5. Which makes sense. We're used to seeing this. When we have y is equal to something times x, this something right over here is going to be our slope. So the equation of the line is y is equal to 0.4x. So let's fill this in. Let's actually do the exercise now. So we had two points, one was the point 0, 0. When x is 0, y is 0. And when x is 5, y is 0.4 times that. So it's y is equal to 2. And we said the equation is y is equal to 0.4 times x. So let's check our answer.