AX plus BY equals AY plus BX? - Answers

Ax + by = ay + bxax - ay + by - bx = 0a(x - y) + b(x - y) = 0(a + b)(x - y) = 0-----------------------------------------------------Another contributor continued:Great !When I saw this question, I was about to say "No it doesn't. Not unless A=B."But "Blackfireweb" developed a much more in-depth analysis. Unfortunately, there was asmall error in his algebra, in the third line, where he collects like terms from the second line.Repeating Blackfireweb's work:Ax + By = Ay + BxA(x - y) + B(y - x) = 0A(x - y) - B(x - y) = 0(A - B) (x - y) = 0-- The equation can only be true for all (x, y) if ( A = B ).-- Otherwise, it's only true at a point where ( x = y ).What this really means is: Whatever numbers you pick for 'A' and 'B', if you write an equationsuch as appears in the question and then massage it a bit, you always find that you havethe equation ( Y = X ), and you realize that we have all been making something complicatedout of something simple.Notice that the "question" doesn't even ask any question. It just makes a statement.To which we reply: "OK. That's the equation ( Y = X ). So what ?"