 In this example, we're going to introduce what we call a mixture problem. Many times, we're dealing with situations with two unknown quantities. For the sake of it, we'll call them x and y. And we know that x and y are mixed together in some regard. Now, if we have two variables, we're going to need two equations in order to solve for a unique solution here. And so these mixture problems naturally turn into systems of linear equations. So let me explain via some examples here. Suppose that 850 tickets are sold for some game and the ticket revenues make the school here $1,100. All right? We know that adult tickets are sold for $1.50 and the children tickets were sold for $1. Could we then determine how many adult tickets were sold and how many children tickets were sold? So like maybe there's like a football game at the local high school and they do charge for tickets maybe. And so we know that our 850 people came and we know that the school made $1,100 in revenue there. So but then we're like, oh, how many adults? How many children? How many students are coming? How many adults are coming? Can we break that down? And so we have two unknowns here. So the first unknown would be how many adult tickets? We'll call that x just for the sake of, you know, we could call it like maybe a and c if we wanted to. So it was called x where x is going to be the number of adult tickets sold at this game. And then y will naturally be the number of children tickets, kid tickets that were sold at this game. So because there was a total of 850 tickets sold, we know that when you combined x and y, you're going to get 850. Combining in this case, of course, means you're adding them together. If you take all the adult tickets and add them with all the children tickets, this will add up to be 850 tickets that much we sold. That's going to be our first equation. And this is always what you get with a mixture problem. You have your two variables which are unknown. If you add the two variables together, you're going to get some quantity, which you should hopefully know. That'll be your first equation, x plus y equals something that something will be given. The next equation, and so this right here is just going to be our sum of the two variables we'll know about that. The next equation you're going to get with these mixture problems is going to have something to do with the weight. That is the two objects being combined together are not equal. That is one might be heavier than the other. For example, adult tickets are more expensive than children tickets. So adult tickets have a rate. Their rate is they cost $1.50 per ticket. So if I take $1.50 and I times that by x, so this is $1.50 per ticket. This is the number of tickets. So when you multiply those together, you're going to get the amount of money from adult ticket sales right there. And so let's make a comment about that. So this combined value right here is the adult, the adult sales. Now if we do a similar thing, if we times the y by one, so children tickets cost $1 per ticket and there were why many tickets sold. So when we multiply those together, this is going to give us the sales for the children. This is the kid sales that they make. And so then if we add up together the adult sales and we add up together the kid sales, this will add up to be the total sales, which is $1,100 in this situation. So this right here is going to be the total sales. And so this is what I mean by this weighted equation. The adult tickets don't cost the same as the children tickets. But when you mix them together, the total amount is going to be $1,100. And so when you consider these things together, this gives us a system of two linear equations with the unknowns x and y. And in which case we want to solve this system of equations. We can solve it using substitution, elimination. The substitution will work really well, but I'm going to do elimination because you know what? You have a y in the first equation. You have a y in the second equation. If I just subtract the equations from each other, they'll cancel out. So we're going to take the second equation 1.5x plus y is equal to $1,100. And I'm going to subtract from it x plus y and 850 right here. So you're going to subtract that. So if you take 1.5x minus x, that's going to give you half at x. The y's will cancel out. Of course, that was the plan there. And then finally, if you take 1,100, take away 850, that will be 250. So then to solve, we'll just divide by 1.5, 0.5. Of course, dividing by 0.5 is the same thing as just times 8 by 2. Because 0.5 is 1.5 and divided by the fraction 1.5 is just times 8 by 2. So you get 2 times 250, which would give us 500 tickets. So x turns out to be 500. That means there were 500 adult tickets sold. Now we have to solve the system, or we have to solve it for y, right? But when we look at the original equation, you have this x plus y equals 850. We can simply just subtract the 500 from the 850, which is going to tell us, let's write it down right here. So we have that y is going to equal 850, take away the 500 adult tickets we sold here. And this is going to give us 350 tickets. So we see that there's a lot more adults attending these games than kids themselves. So maybe mom and dad are coming watching their kids bringing grandma and uncles and aunts and what have you. But turns out we don't actually, comparatively, we don't have a lot of students coming to these games. Maybe this should be a concern for us whatsoever. But based upon what we found here, we see the following. There were 500 adult tickets sold and 350 children tickets sold.