 All right, let's try to do some algebra. Because algebra is a generalized arithmetic, once you know how to do any of the basic arithmetic operations, you can solve algebra problems if you have a conceptual understanding. So let's take a look at some simple algebraic problems we can solve now that we have addition and subtraction. So solve for x, x plus 8 is equal to 15, and let's use three different methods without relying on the so-called rules of algebra. These are things like you can add the same thing to both sides, and you can subtract the same thing from both sides in similar things. You don't actually need these rules. They are irrelevant to the practice of algebra. If you understand arithmetic, you understand algebra automatically. So let's try to approach this problem three different ways. So the first thing we might notice here is we can rely on our definition of subtraction. And for reference, remember that saying that if I have an addition, then by the definition of subtraction, I will immediately have a corresponding subtraction. Instead of using this, I'll make a comparison between our definition of subtraction with the statement that we have. So if I look at my statement, x plus 8 equals 15, and I compare it to this statement here, I see that a and x are playing the same role. So what I'll do is I'll take these a's in both places here, and I'll replace them with an x. I also see that b and 8 are doing the same thing. So I'll replace my b's here and here with 8, and then finally I see that c and 15 are playing the same role. So I'll replace the c's with 15, and I have my definition of subtraction solution. Because I have x plus 8 equals 15, that's what we're starting with, I can immediately say that x is 15 minus 8. Now, as is our convention, anything that's part of our definition we can just do without a comment or explanation. So here I have the subtraction 15 minus 8 as part of my definition. Well, I can just do that subtraction and that gives me my solution. x is equal to 15 minus 8. That's going to be 7. Now, the important thing here is that the solution this problem is going to be all this material that we've shown here in green. We have to include all of that as part of our solution. Well, how about a second solution? Again, not using the rules of algebra. And so here we might use the method of counting up. And so we might consider the problem this way. Since x plus 8 equals 15, then by the commutativity of addition we have 8 plus x equals 15. And so I can count up from 8 some amount until I get to 15. So let's see how we would do that. Again, I might start by counting up by 2 to get me to a number that's easy to work with. So 8 up 2 gets me to 10. I have a target of 15, so I'm going to go 5 more to get to 15. And so how far up did I have to count? Plus 2, plus 5, I had to count up by 7. So that tells me that x is equal to 7. And once again, the part of the solution that is relevant is the part that's shown in green. And finally, I might do a third solution using, for example, a tape diagram. So here I have the equivalence equals of x plus 8 and a 15. So because x plus 8 is equal to 15, then x plus 8 and 15 can be represented by two tapes of equal length. So I'll go ahead and draw that. There's my x plus 8 tape here. There's my 15 tape, also having exactly the same length. And the important thing to note here is this x plus 8 tape is actually x together with 8. So I'll break the tape on the top into two pieces, my x piece and my 8 piece. So I still have x plus 8 up top. I still have 15 on the bottom. And since I just want to find this x piece, I can get rid of these last 8 units. So I'll hack those pieces off, or I'll split them up that way. And if I get rid of those last 8 units, if I do that computation, this 15 minus 8 is going to be my solution, x equal to 7. And again, the essential details are going to be the things shown in green, along with our two diagrams here. And if you want to, you could probably dispense with the first diagram here. The second diagram is sufficient by itself. We've hacked off these 8 pieces here. And so what's left over is going to be the 15 minus 8.