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?

 Jun 4, 2022

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?

 Jun 4, 2022

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