Writing and graphing linear equations

0:00- [Lecturer] At a school spirit store, stickers cost $2 each and pins cost $3 each. A student council member spends exactly $24 on stickers and pins. Let X represent the number of stickers and Y represent the number of pins. Write an equation that represents the number of stickers and pins the student council member bought.

0:22So we'll write that equation first. And then they want us to graph the line that represents the equation. Pause the video and at least think about how you'd approach it. Ideally try to do it yourself. All right, so let's see. Maybe I'll color code this. They say stickers cost $2 each and then the number of stickers is represented by X.

0:41And then they also tell us that pins cost $3 each and Y represents the number of pins. So let's think about the total amount that they are going to spend on stickers. They're $2 each and they're going to buy X stickers. So this is going to be spent on stickers.

1:04So let me write sticker, sticker cost. And then how much are they going to spend on pins? Well, each pin costs $3 and they're buying Y pins. So $3 per pin times Y pins. This will be the pin cost. Pin cost. So how much are they spending in total? Well, it's going to be the sticker cost plus the pin cost.

1:30And they say that the council member spends $24. So the total amount they're spending on both needs to be equal to $24. And just like that, I have written an equation that represents the number of stickers and pins the student council member bought. Now, let's graph the line that represents this equation.

1:50Well, to do that, we could just find some points that are really convenient. If we have two points and connect them, we've got ourselves a line. So let's just think about some X and Y values. The easiest ones for me are, let's imagine we buy zero stickers. So X is zero. Then how many pins are there?

2:08Well, then this equation becomes 3Y is equal to 24. I could write it over here. So if X equals zero, that becomes two times zero plus 3Y is equal to 24. That's just zero. So 3Y is equal to 24, divide both sides by three. We get Y is equal to eight. So we have the 0.08.

2:29So zero, eight. This is the situation where the student is spending all their money on pins. Now let's go with the opposite scenario. What happens if they're buying no pins? Well, then this equation becomes 2X plus three times zero is equal to 24. This part is just zero.

2:51So you get 2X is equal to 24, divide both sides by two, and you get X is equal to 12. So when there's no pins and you spend all the money on stickers, you can buy 12 stickers. So we also have this other, I guess you could say, extreme scenario. So both of these meet the constraints.

3:11They're both on this line. And let's see, I can get my line tool out now, and I can attempt to connect the dots. Let me see how well I can do that. Let me see. Let me get it right on the dot. That's pretty good. And now I can connect the dots. And this is really interesting because every point on this line is actually a situation that might happen.

3:38Now, assuming we can't buy fractional stickers and fractional pins, then it's actually going to be the ones where we have whole numbers of both. So it might be something like that. It might be something like that where we have exactly, in this situation, four pins and six stickers, or we could have two pins and nine stickers.

4:04And you can validate that that actually works. Two pins and nine stickers is going to be two pins and nine stickers. Nine stickers is going to cost $18. Two pins is $6. You add them up, you get to 24. So the line represents all of the different combinations of pins and stickers that work.

4:21And assuming we can't buy fractions of pins and fractions of stickers, the points where we actually hit at whole numbers of both, those are the actual combinations that work.