Carl Horowitz

**University of Michigan**

Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

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A system of linear equations is two or more equations that contain the same variables. A solutions to a system of equations are the point where the lines intersect. There are four methods to solving **systems of linear equations**: graphing, substitution, elimination and matrices. Solving systems of equations first shows up in Algebra I, but more complex applications occur in Algebra II.

So often times we're given situations where we're trying to figure out the number of possible outcomes for a set of information, okay? And one example that I want to start with is just a going to a sandwich shop and dealing with the lunch that we're receiving. So if you buy a sandwich you can get a choice of a soda or tea and a side of fries, chips, slaw or salad. and we're trying to figure out the number of different combinations that you can get for this particular problem.

So one of the easiest ways to start with this kind of problem is to just make a what's called a tree diagram, okay? And it's really like what it sounds like it just sort of connecting branches of a tree. So what we end up with is we start with our sandwich, we start at one point. From there we can either get our soda or our tea. Okay? So that takes us through one decision that we have to make. Then once we end up at that point we then can choose our other side. So if we got our soda, we could go with our fries, side chips or slaw or our salad. Okay? We could also do those same different combination with our tea. We could get those fries, chips, slaw or salad and then all we have to do is once we go through all those things is count the number of endpoints that we have. So with our soda there were 4 different options we could get. With our tea there were 4 different options that we could get for a total of 8 different things, okay.

So in general the tree diagram is a good way to sort of start organizing your data but once you start dealing with a lot more ingredients. So say we had you know different toppings on our sandwich or many different deserts or who knows what, you're going to start getting a lot of branches and it's going to become a little bit overwhelming. It will always get you through it but it's not always going to become the most efficient thing, okay?

So there's also what we can do is called the fundamental counting principle and in general I'm not a huge fan of definitions but I've done this one up here just so you can actually see how it works, okay? So what it says if there are m ways for one event to occur and n ways for another, then there are m times n ways for both to occur. So really all you have to do is take the possible numbers of one outcome times the possible numbers of the other and that's going to be your answer. Going back to our sandwich. There were 2 drinks, there were 4 solids. So the number of possible events is just 2 times 4 which is 8, okay? So we're often times able to do a tree diagram but more often than not and it's going to be significantly easier we can just use the fundamental counting principle, multiply our individual components together to get our answer.