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!
+23254 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.
