Combining like terms example

We have a hairy looking expression here and your goal is to try to simplify it as much as you can. I'll give you a little bit of time to do it Let's just think about it step by step It might help if you were actually to reorder the terms in this expression So let me put all the "x" terms first so I have"5x" that's that term minus 2x plus 7y plus 3y then I have "+ 8z" And then I have minus Z and the last term that I haven't included yet is "+5". Right over here Now let's just think it through. If I have 5 x and I were to take away 2 x how many x would I be left with? Well I would be left with 3 x. That's true of anything. There's not same fancy algebraic magic going on. 5 of anything minus two of that same thing You're going to be left of 3 of that same thing In this case that "thing" is x So this is going to simplify to 3x Now in a lot of algebra classes you will hear people say, "Oh, well the coefficient on 5x is 5 and the coefficient here is negative two and we have to add the coefficient (let me write that word down) these right over here are the coefficient they're the number that your mutiplying the variable by So the 5 or the negative 2 in this case you could just so Oh I can just add the coefficient and that's ok. There's nothing wrong with that but I really want to emphasize there's a very common sense intuition here You have 5 of something you take away two of that something you are left with three of that something you have to make sure that your adding and subtracting the same thing here we're dealing with x's so we can take 5x and take away 2x we can't think about merging the x and the y at least, not in any simple way right now because frankly that would make any intuitive sense now let's think about the y if I have 7 of something and I were to add 3 more of that something then I would have 10 of that something so this part right over here is going to simplify to 10 y once again you could say the coefficient on 7y is 7 the coefficient on 3y is 3 we added the coefficients (7+ 3) to get to 10y but I really want to emphasize the intuition here If you have 7 of something you add another 3 of something, you have 10 of something now let's look at the Z if I've got 8 of something and I take away one of them I'm going to have 7 of that something that is 7z and you might say hey what was the coefficient here on the -Z I don't see any number out front of the z. Well implicitly I could have put a one here. It's exactly the same thing. Subtract a Z is the same thing as subtracting 1z the word 1z strikes a part of my brain because I have very young children but that's a different type of 1z and then you can see yeah you definetly did add the two coefficients The 8 and the -1 but once again common sense tells you that if you have eight of something and you take away one of them you have 7 of that something and then finally you have a plus 5 so we're done this simplified to 3x plus 10 y plus 7z plus 5