Setting up systems of linear equations example

In this video, we're going to get a little bit of practice setting up systems of linear equations based on a word problem. We're not actually going to end up solving it; you can do that if you like, just for kicks. But really, we're going to just focus on setting it up.

So here, we're told 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 total.

Let d be the number of kilograms of dark roast beans she buys and l be the number of kilograms of light roast beans she buys. All right, so based on this information that we've been given, see if you can pause this video and set up a system of equations. It's going to have two equations with two unknowns, d and l, that in theory we could solve to figure out the right number of kilograms of dark roast beans and light roast beans that Lauren should use. So pause this video and try to work that out.

All right, now let's do it together. What I'm going to do is I'm going to underline. So let's see, we know that d is dark roast beans and l is the number of kilograms of light roast beans. Then they tell us here, they say she needs 80 kilograms of beans in total. So what we could say is, hey, the number of kilograms of dark roast beans plus the number of kilograms of light roast beans needs to be equal to 80 kilograms.

So, the number of kilograms of dark roast beans plus the number of kilograms of light roast beans—I'm having trouble saying light roast beans—well, this what I just underlined here, it says needs to be 80 kilograms in total. So that needs to be 80. This number of kilograms plus this number of kilograms is going to be equal to your total number of kilograms.

All right, so I have one equation with two unknowns. Let's see if we can get another one. Next, they say dark roast beans cost three dollars per kilogram, light roast beans cost two dollars per kilogram, and she wants to spend 220 total. So what I just underlined in this aquamarine color, we can set up another equation with. If you haven't already set up your system of equations, see if you can now do that. See if you can set up the second equation; pause the video.

All right, well, the way to think about it is we just have to have an expression for how much does she spend on dark roast beans and how much does she spend on light roast beans. Then we need to add those two together, and that needs to be equal to two hundred and twenty dollars because that's how much she wants to spend in total.

So how much does she spend on dark roast beans? Well, it's going to be the number of kilograms of dark roast beans that she buys, and it says that it costs three dollars per kilogram, so we're going to multiply it by three. Three dollars per kilogram times the number of kilograms of dark roast beans—this is how much she spends on dark roast beans.

And so how much is she going to spend on light roast beans? Well, she buys l kilograms of light roast beans—they told us that there—and they cost two dollars per kilogram. So two dollars per kilogram times the number of kilograms—this is how much she spends on light roast beans.

So you add how much she spends on dark roast to how much she spends on light roast, and so this is going to be 220 dollars in total. There you have it! We have our two equations with two unknowns.

So now we could go and solve it, but you can do that outside of this video. The whole point of this video is to understand how to construct these based on the constraints, based on the information that we see in this. So typically, when you're trying to set these up, there's often a sentence or two that will focus on one equation.

So this first one is saying, hey, the kilograms—let's add those up for the total number of kilograms. And then there's another sentence or two that will focus on, in this case, some other equation—in this case, it's the price. So, the price of the dark plus the price of the light is going to be equal to the total amount she wants to spend.