I pick two whole numbers $x$ and $y$ between $1$ and $10$ inclusive (not necessarily distinct). My friend picks two numbers $x -4$ and $2y-1$. 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?
All work shown would be appreciated!
If both x and y must be whole numbers, then: (x - 4)(2y - 1) = xy + 1
Multiplying out ---> 2xy - x - 8y + 4 = xy + 1
Rearranging terms ---> 2xy - xy - 8y = x - 3
xy - 8y = x - 3
Factoring ---> y(x - 8) = x - 3
y = (x - 3)/(x - 8)
If you test the number between 1 and 10 for x, you can get these whole number answers: (3,0), (7,-4), and (9,6).
Of these, only the answer x = 9 and y = 6 have answers between 1 and 10 for both x and y.