jmc

Question

For any two real numbers x and y, define x y = ax + by + cxy,where a, b, and c are constants. It is known that 1 2 = 3, 2 3 = 4, and there is a non-zero real number d such that x d = x for any real number x. What is the value of d?

Accepted Answer

Setting x = 0 in x d = x, we get 0 d = 0, so bd = 0. Since d 0, b = 0, and x y = ax + cxy.From 1 2 = 3, a + 2c = 3. From 2 3 = 4, 2a + 6c = 4. Solving, we find a = 5 and c = -1. Then d satisfies x = x d = 5x - dxfor any real number x. This implies 5 - d = 1, so d = 4.