Combining like terms with negative coefficients

now we have a very very very hairy expression and once again I'm going to see if you can simplify this and I'll give you a little time to do it so this one is even crazier than the last few we've looked at we've got wise and XY z-- and X Squared's and X's and well more just X YS and Y Squared's and on and on and on and there will be a temptation because you see Y here and a y here to say oh maybe I can add this negative 3y plus this for X Y somehow since I see Y and a y but the important thing to realize here is that a y is different than an XY thinking about if they were numbers if Y was 3 and then X was a 2 then a Y would be a 3 while an XY would have been a 6 and a y is very different than a Y squared once again if the Y took on the value 3 then the Y squared would be the value 9 so even though you see the same letter here they aren't the same I guess you can't you cannot add these two or subtract these two terms so Y is different than a Y squared is different than an X Y now with that said let's see if we can if there is anything that we can simplify so first let's think about the Y terms so you have a negative 3y there do we have any more Y terms yes we do we have this 2y right over there so I'll just write it out I'll just reorder it so we have negative negative 3y plus 2i plus 2i now let's think about and we can I'm just going in an arbitrary order but since our next term is an XY term let's think about all of the XY terms so we have plus 4xy right over here so let me just write down I'm just reordering the whole expression plus 4xy and then I have minus 4xy right over here so minus 4xy then let's go to the x squared terms I have negative 2 times x squared or minus 2 x squared so let's look at this so I have minus 2 x squared do I have any other X Squared's yes I do I have this 3x squared right over there so plus 3 X Squared's and then let's see - why I have an X term right over here and that actually looks like the only X term so that's plus 2x and then I only have one y squared term I'll circle that in orange so plus plus y squared so all I have done is I've reordered the statement and I've color-coded it based on the type of term we have and now it should be a little bit simpler so let's try it out if I have negative three of something plus two of that something what do I have or another way to say it if I have two of something and I subtract three of that what am I left with well I'm left with negative one of that something so I could write negative I could write negative 1y or I could just write negative Y and another way to think about it but I like to think about it intuitively more is what's the coefficient here it is negative 3 what's the coefficient here it's 2 we're obviously both dealing they're both Y terms not X Y terms not Y squared terms just Y and so negative 3 plus 2 is negative 1 or negative 1 Y is the same thing as negative Y so those simplify to this right over here now let's look at the X Y terms if I have 4 of this for XYZ and I were to take away 4 x wise how many XY is am I left with well I'm left with no x wise or you could say add the coefficients 4 plus negative 4 gives you 0 x wise either way these two cancel out if I for something and I take away those 4 of that something I'm left with none of them and so I'm left with no X wise and then I have right over here I could have written 0 X Y but that wouldn't you know that that seems unnecessary then right over here I have my x squared terms negative 2 plus 3 is 1 or another way of saying if I have three X Squared's and I were to take away two of those X Squared's I'm left with 1 x squared so this right over here simplifies to 1 x squared or I could literally just write x squared 1 x squared is the same thing as x squared so plus x squared and then these there's nothing really left to simplify so plus 2 X plus 2 X plus y squared plus plus y-squared and we're done and obviously you might have got an answer in some other order but the order in which I write these terms don't matter just matters that you were able to simplify it to these four terms