Slope and intercept meaning from a table | Linear equations & graphs | Algebra I | Khan Academy

We're told that Felipe feeds his dog the same amount every day from a large bag of dog food. Two weeks after initially opening the bag, he decided to start weighing how much food remained in the bag on a weekly basis. Here's some of his data: So we see after 14 days, there's 14 kilograms remaining. Then after another seven days passed by, so now we're 21 days from the beginning, there's only 10 and a half kilograms left. Then after 28 days, there are seven kilograms left.

All right, so we're going to try to use this data to start answering some interesting questions, and maybe we'll also try to visualize it with a graph. So the first thing that we might try to tackle is, well, how much food was in the bag to begin with? If we assume that he's using the same amount of food every week, so pause this video and see if you can figure that out: How much food was in the bag to begin with if we assume that Felipe is feeding his dog the same amount every week?

Okay, now, there are several ways to do this, but to help us visualize this, let me see if I can graph the data that we have and then see what would happen as we approach the beginning of this, of what's going on here, the dog feeding, and maybe as we go to the end as well. So let's see, this is my x-axis. This is my y-axis. I'm going to make the x-axis measure the passage of the days, so number of days on the x-axis. On the y-axis, I'm going to measure food remaining, and that is in kilograms.

And let's see, it looks like maybe if my scale goes up to... let's make this 5, 10... 15, 20, and then 25. I can make it a little higher, 25, I think this will be sufficient. And then we want to go up to 28 days, and it looks like they're measuring everything on a weekly basis, so let's say that this is 7, 14, 21, and then 28. They gave us some data points. So after 14 days, there's 14 kilograms remaining—so, 14 days, there's 14 kilograms remaining right over there.

After 21 days, there's 10 and a half kilograms remaining—21 days, 10 and a half is right about there. After 28 days, there's 7 kilograms remaining—so after 28 days, 7 kilograms. We're assuming the rate of the dog food usage is the same; that he's feeding his dog the same amount every week, and so this would describe a line. The rate is going to be the slope of that line.

Then, if we can plot this line, if we know where that line intersects the x and y axes, we might be able to figure out some other things. So, actually, let me draw a line here. Let me see if I can use this little line tool to connect the dots in a reasonable way. So let's say it looks something like that—that's our line that will describe how quickly he is using his dog food.

So let me make sure that this dot should be on the line as well. Now let's try to answer that first question and think about how we might do it: How much food was in the bag to begin with? So what point here represents how much food was in the bag to begin with? Well, that's the amount of food remaining at day zero, at the beginning of this. So that would be this point right over here; it would describe how much food was in the bag to begin with.

This would be the y-intercept. The y-intercept is when our x-value is equal to zero; what is our y-value? When we just look at the graph, it looks like it's a little bit over 20. But we could find the exact value by thinking about the slope, which is thinking about the rate at which the dog food is being depleted. We can see that every week, every week that goes by—or every seven days that goes by—it looks like we use three and a half kilograms.

Or, another way to think about it is every two weeks it looks like we use an entire kilogram. So let me put it this way: When we go plus 14 days, plus 14 days, it looks like we use up—or the food remaining goes down by—negative 7 kilograms. So, if we want to figure out this exact value, we just have to reverse things. If we are going back 14 days, then we're going to go up 7 kilograms.

So, if we were at 14 up 7 kilograms, this right over here is the point (0, 21). So how much food did Felipe start with in the bag? 21 kilograms. And we got that from the y-intercept. Now another question is: How much is Felipe feeding his dog every day? Pause this video and see if you can figure that out.

Well, we know every 14 days he's feeding the dog seven kilograms. So, one way to think about it is—and we're really looking at the slope here to figure out the rate at which he's feeding his dog—so the slope is equal to our change in the y: so negative seven kilograms over our change in the x—in every 14 days. So 7 over 14 is the same thing as one-half.

So this is equal to negative one-half of a kilogram per day. So this tells us that every day the food remaining is going down half a kilogram. So that means he's feeding his dog—assuming his dog is eating the food and finishing it—that his dog is eating half a kilogram a day.

If we wanted to ask another question: How many days will the bag last? How would you think about that? We know it's going to be out here someplace if we just continue that line, because this point right over here where our line intersects the x-axis, that would be our x-intercept. That is the x value when our y value is zero, and our y is the amount of food remaining.

So we want to know what day do we have no food remaining, and we could try to estimate it, or we could figure it out exactly. We know that every 14 days we use up seven kilograms, so if we are at 7 as we are right over here, and we go 14 days in the future, we should use up the remaining contents of the bag.

So plus 14 days, we're going to use up the remaining 7 kilograms and so this should happen 14 days after the 28th day. So this is going to be the 42nd day.