 This theory is going to deal with systems of equations and their applications. As a general rule, we need to write equations that represent the same thing in each part of the equation. So it's like apples and apples. We're not going to add apples and oranges. If we know a total number of items, then the equation needs to have the number of items in each term. If we know a total value, then the equation needs to have a value in each term. Two families go to the movie together. One family purchases four children's tickets and two adult tickets. The other family purchases three children's tickets and two adult tickets for $24. How much of each type of ticket? Well, first we have to figure out what our unknowns are. And they're asking us for type of ticket. So that would be what our unknowns are going to represent. So let's let C equal children. That way it'll make sense when we get done. And let's let A equal adult. So if we want to write our system, we need to then go back and read the problem. One family purchases four children's tickets and two adult tickets. And then we have a total of $28. Another family gets three children's tickets and two adult tickets. And then that gives them a total of $24. So we're going to take each family, since that's the two different things that we're dealing with, we're going to take each family and let them represent one equation by themselves. So for C, I'm going to do the first one in all red, plus 2A, which is our adult, and that was equal to $28. And then the other family was three children plus two adults, and that was only $24. So now we have a system here that we could easily solve by elimination, because our A's have the same coefficient, we just need to multiply by a negative. So if I come through and multiply the bottom equation by a negative one, just because they're smaller numbers, then I'm going to have a new system of 4C plus 2A equal to 28. And my new equation is going to be negative 3C when I distribute, and negative 2A equal to, and don't forget to go all the way through, negative 24. Now I'm ready to add my two equations. And when I add my two equations, I get 4C minus 3C, so that's just 1C. My A's cancel out, and we're left with 28 minus 24, which is 4. So now we know C, and we just have to find A. Again, remember, you just plug it back into one of your original equations. I'm going to plug it back into the bottom equation only because it was a little bit smaller. The numbers were a little bit smaller. That's the only reason why. So 3 times my C, which is 4, and then plus 2A is equal to that 24. Well, 3 times 4 is 12 plus 2A equal to 24. And then I'm going to subtract my 12 from both sides, and 2A is going to be equal to 12. And if you divide by 2, then we find out that A is equal to 6. So if I write a nice little sentence, I would say that children, a child ticket is $4, and adult ticket is $6. So let's look at another example. Lori makes her own trail mix with her natural foods grocery stores. She makes a Spanish cashews costing $9 per pound, and almonds costing $7.50 per pound. If a three-pound bag costs Lori $24, how many pounds of Spanish cashews and how many pounds of almonds are included in the mixture? We want to fill in the table here. The white part is not the green one. This table will help us set up our equations. So we need to know the amount. We have our items here, so the cashews and the almonds. I already set that one up for us. So we want to know how many pounds of cashews. Well, that's what the question is asking. How many pounds of cashews and how many pounds of almonds? So those are our unknowns. So we're going to say that there's C pounds of cashews to keep it straight, and there are A pounds of almonds. But we do know that the total pounds, here's another pound, is 3. So really we have an equation here. We'll write it across, but up and down is C plus A is equal to 3. Now we need to talk about values. Well, the value of the cashews up here says that it's $9 per pound. So we have $9. And the almonds cost $7.50 per pound. So that would be 7.5. We don't need to have a value of the total yet, because that is in the next column. So now we have a total value that we need to think about. The total value here is the amount times the value is equal to the total value. So we have 9C, since we used to put the number first, and we have 7.50A. And then we have the total that we're spending, or the total value of that bag is $24. So again, we have two equations. This is our second equation. Up and down, we can write it also left and right. So C plus A is equal to 3. And 9C plus 7.50A is equal to 24, which I have right here. So now we have options. We could multiply the top equation. Probably the 9 would be the easier one to do by negative 9. And we could use elimination. Or we could solve A or C for A or C, and use substitution. The last problem we did by elimination. So let's try this one by substitution. So A looks like it would be nice to distribute into, because 9 would distribute nicely. So I'm going to solve for A. And this equation, I'm going to subtract C from both sides. So that gives me nothing on the left-hand side but A. And then 3 minus C on the right-hand side. So rewriting this bottom equation then for substitution, I have 7.5. I dropped the 0. It's the same thing as 7.50. Plus 9, but now instead of A, I'm going to write 3 minus C. And then that's equal to R24. Alright, doing our work then. 7.5 C plus, and then we're going to distribute here. And we're going to get 27. And distribute here and get minus 9 C equal to R24. So 7.5 minus 9 and negative 1.5 C. And then plus to 27 is equal to the 24. So track the 27 from both sides. Negative 1.5 C is going to be equal to negative 3. And if we divide by negative 1.5, that should be 2. So we now know that cashews cost $2. But we still don't know what our almonds cost. But remember we have this nice equation up here where A is equal to 3 minus C. And we can plug and chug again. So we have A equal to 3 minus, but now we know C to be 2. So that tells us that A is equal to 1. So putting it in a sentence then, cashews are 2 pounds and almonds 1 pound. To make the 3 pound mixture.