 Okay, again, there is no note sheet for this section, so you're going to copy this or design this proof in your notes. So make sure you have your notes handy, ready to go. And we're going to start by writing down the given and the proof. Two of the five steps, essential steps to a good proof are writing down the given and the proof. So we're going to start with writing the given and the proof. The given is the hypothesis of the if-then statement, it's the if portion of the if-then statement. So we've written down the given. Next we'll write down the proof, you're writing this on your paper. The proof is the conclusion of the if-then statement, the then part of the if-then statement. And so we're going to prove that x equals 12. So we've written down the given and the proof of the algebraic proof. Because it's an algebraic proof, there is no diagram. Every geometric proof you will design will have a diagram, but with algebra we don't have diagrams, so there will be no diagram to copy down. A place to start, every proof is with the given. So we will again write down the given statement. So statements are compartmentalized, are boxed off, and then for every statement that we make we need to have a reason. And the reason for this statement is because it was given to us. The first thing you would do to solve this algebraic problem would be to use the distributive property. So I am going to distribute the 4 across the 2x and the 3. And I will get 8x minus 12 equals 84. I will box that statement off. And the reason that I could move from this first statement to the second statement is the distributive property. So I make a note that the reason I can go from the first statement to the second statement is because I applied the distributive property. The next thing I would do to solve this problem is to add 12 to both sides. And if you need to make a notation in your notes that you are adding 12 to both sides you may do that. The new statement will then be that 8x equals 96. We will box that off. And the reason we could make that statement is the addition property of equality. It says as long as you add the same thing to both sides you haven't changed the equation. So we will call that the addition property. The next step in solving this equation or algebra problem would be to divide both sides by 8. And if you need to make a notation that's great. When I divide both sides by 8 I get that x equals 12. I box off my statement. And the reason I can move from this step here to this step here is because I applied the division property of equality division. Spelling is important. The division property and which states as long as I divide the same number on both sides of an equation I have not changed the equation. So I still have an equation. I have now proved that if 4 times the quantity 2x minus 3 equals 84 then x equals 12. And I know you are all saying but I could do that before. Yes you could. The reason we start with this kind of proof is because you already know how to do the math. Now we are giving you the skills to design the flow proof. And I don't know that you have ever before given a reason for every single step of an algebra problem that you have solved. But now this is the design of a good flow proof with statements and reasons. Just as an aside, back in the day, back in the something hundreds, geometers, people who did proofs used to end their proofs with QED. You can ask your teacher about that.