Interpreting points in context of graphs of systems

We're told that Lauren uses a blend of dark roast beans and light roast beans to make coffee at her cafe. She needs 80 kilograms of beans in total for her next order. Dark roast beans cost three dollars per kilogram, light roast beans cost two dollars per kilogram, and she wants to spend 220 dollars in total.

Then they tell us, here's a graph that shows a system of equations for this scenario, where x is the number of kilograms of dark roast beans she buys and y is the number of kilograms of light roast beans she buys.

All right, let me scroll down so we can take a look at this. And so sure enough, so this blue line—and I'll write it out in blue—this x is the number of kilograms of dark roast beans; y is the number of kilograms of light roast beans. She wants to buy a total of 80 kilograms; that's what they told us up here. We can go back to look at that; she needs, underlined in blue, she needs 80 kilograms of beans.

So that constraint that the sum of the kilograms of dark and light is equal to 80. That's represented by this equation, and if we were to graph it, that is this blue line right over here.

Then this other constraint: 3x—well, the dark roast beans cost three dollars per kilogram, so 3x is how much she spends on dark roast. 2y is how much she spends on light roast because it's two dollars per kilogram, and 220 is the amount that she spends in total. They tell us that up here: dark roast beans cost three dollars per kilogram, light roast beans cost two dollars per kilogram, and she wants to spend 220 dollars.

So this equation is another way of expressing what I just underlined up here in green, and the green line shows all of the x-y combinations that would match these constraints.

And so now let's do something interesting. They've labeled some points here: point C, D, F, and E. We're going to think about what each of these points represents.

So for example, this point C, that is on the green line but it sits above the blue line. What does this tell us? What does this point C represent? Pause this video and think about it.

Well, if we're on the green line, that means that the amount that she spends on dark roast plus the amount that she spends on light roast is adding up to exactly 220 dollars. So she's definitely spending 220 dollars at C. But how many total kilograms is she using? Well, the fact that for this given x we're sitting above the line means that she's not using exactly 80 kilograms.

We can see that over here; she's using, looks like, 10 kilograms of dark and it looks like something like 95 kilograms of light. If you were to add those two points together, it looks like she's using something closer to 105 kilograms.

So point C is a situation where she is spending exactly 220 dollars, but she's using more than 80 kilograms because it's not meeting this second constraint; it's sitting above that line.

Now let's think about point D. What does that represent? Pause the video and try to figure that out.

Well, because we sit on the blue line, that means that we are meeting this constraint that the kilograms of dark and light combined is equal to 80 kilograms. So she's using exactly 80 kilograms here. But what about her spending?

Well, because this point lies below the green line, that tells us that we are spending less than 220 dollars. We could even try it out: 3 times 20 plus 2 times 60 is what? 60 plus 120 is 180. And so this is a point where we're meeting this constraint, but we're not meeting this constraint; we're underspending right over here.

Now what about point F? Well, point F sits below both of these lines, so pause your video and think about what that means.

Well, if we're sitting below both of these lines, that means that neither are we spending 220 nor are we using 80 kilograms. You could see that if you actually look at the numbers; you don't have to do this, but this is just to make you feel good about it. It looks like she is using 30 kilograms of dark and 30 kilograms of light, so in total she is using—so this is a situation where she's using 60 kilograms in total, not 80.

And so that's why we're not sitting on this blue line. If you look at how much she's spending: she has 30 kilograms of each, so 3 times 30 plus 2 times 30; that's going to be 90 plus 60; that's also less than 220. And so that's why we see this point is below these lines.

And then last but not least, what does point E represent? Well, point E sits on both of these lines, so that means that it meets both of these constraints. This is a situation where she is spending exactly 220, and the total number of kilograms she's using of dark and light is exactly 80.

And so, if we wanted to say, "Hey, what combination of dark and light would she need in order to meet both constraints?" E represents that—the intersection of these two lines.