I pick two whole numbers x and y between 1 and 10 inclusive (not necessarily distinct). My friend picks two numbers x -4 and 3y-1. If the product of my friend's numbers is one greater than the product of my numbers, then what is the product of my numbers?
We have the following equation: \(xy + 1 = (x-4)(3y-1)\)
SImplifying the right hand side and isolating the constant gives us: \(-3 = 2xy - x - 12y\)
From here, we can factor out \(2y\) from the right-hand side of the equation. This gives us: \(-3 = 2y(x-6) - x\).
Now, notice how if we add 6 to both sides of the equation, we can factor in terms of \((x - 6)\): \(3 = (2y - 1)(x-6)\).
Now, we need 1 term to equal 3, and the other to equal 1. There are 2 ways to do this, but only 1 will satisfy the question.
Can you take it from here?