 So we've looked at some of the aspects of trigonometry, geometry. We've looked at some coordinate geometry and some simple plane geometry with right angles and different aspects of geometry. We've talked about some of the past history of geometry and why geometry is one of the most important branches of mathematics. Now what we're going to do is apply some of the stuff that we've learned to specifically problems involving proofs or just basically problem sets where they give you a certain set of parameters and they ask you to solve for certain variables or just prove a certain function or certain relationship. Keep in mind that when we're doing simple algebra there are two different types of questions that we're asking you. One is simplify and the other one was solve. Now the difference between simplify and solve is with solve in the question there is an equal sign. So for example they would ask you here something like 2x plus 7x is equal to 18 and you would have to solve for the variable x. So 2x plus 7x is the same thing as 2 apples plus 7 apples is 9 apples. 9x is equal to 18 and you divide by 9 to get your variable by itself. So x is equal to 2. With simplify the question is going to be missing the equal sign. It's going to say 2x plus 7 and all that we say is simplify. So 2x plus 7 is going to be 9x and that's your answer. There is no solution for x here. This is a general term. This is simple algebra with equations and just simplification. The same type of layout works when they're asking you questions in a problem form. So what we have here is if they ask you there's two different ways that can ask you problems. One of them is just simply asking you a problem and the other one is asking you to prove a certain relationship or a certain problem. With problems the way they lay them out is basically English. They use words to lay out a problem and they ask you to solve for a certain variable. For example a question would be like John gets $12 an hour and if he works 40 hours then how much does 40 hours per week how much would he get paid for a five day week? Now this is just a straightforward problem. What you would do is he works 40 hours per week he gets paid $12 an hour and he works for five weeks. So all you would do is go 12 times 4 that tells you how much he's going to make in a week and that gives you how much money he's going to be making in five weeks. Proofs work differently. Proofs are general solutions for a certain type of problem. The way it works is proofs are way more powerful because they lay out the certain format that you're going to be solving certain types of problems and it's going to work exactly the same way for every type of problem they give you. So proofs are general. There is no numbers really involved. It's going to be all variables like X and Y, Q and W. So another form is going to be when they ask you to prove this side of the equation equals this side of the equation and there are going to be different variables on each side. So keep this in mind when we're dealing with problems with questions involving problems and when we're dealing with proofs. Thank you.