Modeling with multiple variables: Taco stand | Modeling | Algebra 2 | Khan Academy

We're told a certain taco stand sells t tacos per day for a net profit of 300. Each taco costs c dollars to make and is sold for p dollars. Write an equation that relates t, c, and p, so pause this video and see if you can do that.

All right, now let's work this together. So let's just remind ourselves what's going on here. We have the number of tacos sold per day is t, so t equals the number of tacos sold per day. They tell us net profit of 300, so we could say 300 is equal to net profit. They tell us that each taco costs c dollars to make, so c is equal to cost per taco. And then p is what each taco is sold for; right over there is sold for p dollars, so p is equal to the price per taco.

So how do we figure out what the net profit is going to be? Well, we could write it this way: net profit is going to be equal to the total amount of money that you bring in minus the total amount of money that you have to spend. So what is going to be the total amount that you bring in? What's going to be the amount you sell times what's going to be the number of tacos you sell times the price per taco? So t times p, this is how much money you're bringing in.

But of course, you have to also subtract out your cost. Well, what's going to be your cost? What's going to be the number of tacos times the cost per taco? That's how much we're going to spend, so the number of tacos times the cost per taco. Now, here I've written everything as variables. Well, I wrote net profit out, but they told us that net profit is equal to 300. So I could write it like this: 300 is equal to t times p, I'll do those same colors, t times p minus t times c minus t times c.

Now, it's completely possible that they didn't give us net profit as 300 per day; instead, they said that is one of the variables and they gave one of the other variables. And that would have been okay. We could have used the same logic. Whatever they didn't give us, we could have set up as a variable, and whatever they did give us, maybe they gave us the price per taco; we could have put that as a given number.

Now, I know what some of you all are thinking. You didn't get exactly this. You might have thought about it the other way. You might have thought about it as the profit is going to be equal to the number of tacos times the profit per taco. And what's going to be the profit per taco? Well, that's going to be how much each taco is sold for minus how much each taco costs. And this also would have been completely credible.

And if you look at these two, they're actually algebraically equivalent. If you factor out t here, you get this expression on the right.