 In this video, we're going to do another mixture problem, which we will be literally mixing materials this time. So imagine we're going to construct, we're going to make an alcohol solution. We want that alcohol solution to be 30% alcohol, right? It's 7% water. And we're going to accomplish making this 30% alcohol solution by combining 20% alcohol solution, which we already have in hand. And we also have 50% alcohol solution. So if we mix them together, we can make an alcohol solution of 30%. And let's say that we want to make 12 gallons of the 30% alcohol solution. And so this is a very classic mixture problem. And we have to do quantities we have to decide upon. So we have some amount of the 20% and some amount of the 50% that we have to add together. So we're going to let x here represent the number of gallons of the 20% solution. And we're going to combine this with y, which is going to equal some number of gallons of the 50% solution. We want to add those together. So what we know is that when we add together x and y, this will be a total of 12 gallons. So if we mix together the complete solutions, we get this total sum right here. This should add up to be 12 gallons. Keeping track of units can be very useful here. Now, the next thing we're going to do is we want to keep track of, like I said, with a mixture problem, there's always this weighted equation because the different quantities have different weights associated to them. Like when we add together the 20% alcohol solution, 20% of its volume is pure alcohol. And so let's think about for a moment what this would look like. What does this mean, 0.2 times x? Well, x is the volume of the solution. 0.2x would be 20% of the solution, which is pure alcohol. So this is measuring right here. This is the number of gallons of alcohol in the 20% solution. I should say in the 20% solution. If we add that to, say, 0.5 times y, again, thinking quantitatively what this is, this is going to be the number of gallons of the alcohol, of pure alcohol, in the 50% solution. Well, when we add these together, this thing should equal, if we consider the units here, this should be the number of gallons of alcohol that's going to be in the 30% solution. And so then we're tempted to say something like, well, OK, how many, what's that going to be? There's going to be 30%. And a common mistake on these mixture problems is the students just stop right here. They say 0.2x plus 0.5y equals 30%, which if you stop and think about that, that doesn't make any sense, because this right here is adding together a number of gallons. This is a number of gallons. And gallons plus gallons equals percent? No, no, no, no, no, no. This should be gallons. This should be some type of measurement of gallons, because again, we're trying to measure the number of gallons of alcohol in the 30%. Well, you're going to take 30% of some volume, right? It's going to be 30% of 12, because there's 12 gallons of 30% solution that we're trying to create here, and we times that by 30%, OK? And so this gives us then our system of equations right here. So we get x plus y equals 12. If you don't really like the decimals here, we can times the second equation by 10, which basically has the effect of moving all the decimals over by 1. So you're going to get 2x plus 5y, and then you're going to get 3 times 12, which is 36. And so this is the system of equations that we want to solve. We can do this by substitution or elimination, whichever you prefer. I'm going to do this one by substitution, where you can solve for y in the first equation. You get y is equal to 12 minus x. Substitute this into the second equation down below. So we're going to get 2x plus 5 times 12 minus x is equal to 36. Distributing the 5 there, we're going to end up with 2x plus 60 minus 5x is equal to 36, combine some like terms. The 2x combines with the negative 5x there, to give us a negative 3x. We're going to move the 62 to the other side, so we get 36 minus 60, which is going to give us a negative 24. And then dividing both sides by negative three, we get x equals negative 24x over negative three, which is going to give us eight gallons of the 20% solution. Now, since we solved this by substitution, just take this eight gallons and go back in here. So we get 12 minus eight, which is going to be four gallons. Y equals four gallons of solution right there. And so that then gives us the answer we were looking for. So we see that what we need to do is the following. If we mix eight gallons of 20% solution with four gallons of the 50% solution, we will have 12 gallons of 30% solution. And so a chemist could actually create a mixture of solutions by taking a low concentration and combining it with a high concentration and knowing how many gallons to mix together comes down to solving a system of two linear equations with two variables.