AX plus BY equals AY plus BX?

Answer 1

Ax + by = ay + bx

ax - ay + by - bx = 0

a(x - y) + b(x - y) = 0

(a + b)(x - y) = 0

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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 a

small error in his algebra, in the third line, where he collects like terms from the second line.

Repeating Blackfireweb's work:

Ax + By = Ay + Bx

A(x - y) + B(y - x) = 0

A(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 equation

such as appears in the question and then massage it a bit, you always find that you have

the equation ( Y = X ), and you realize that we have all been making something complicated

out 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 ?"