PLEASE HELP

Question

How many ordered pairs of positive integers (x,y) satisfy the equation x=4-x/y² -x?

 

Pls help Thanks so much<3  

Answer 1

If the equation is  x = (4-x)/(y²-x)  then  solving for  y²  yields  

 

x(y²-x) = 4 - x

xy² - x² = 4 - x

xy² = x² - x + 4

y² = (x² - x + 4)/x

y² = x - 1 + (4/x)  

 

If x is a positive integer then the right hand side will only be an integer if x = 1, 2 or 4.  

 

x = 1  ==>  y² = 4  ==>  y = 2  since y must be a positive integer which checks since  1 = 3/3

x = 2  ==>  y² = 3  ==>  y is not a positive integer

x = 4  ==>  y² = 4  ==>  y = 2.  However, this is an extraneous solution since plugging in x = 4 and y = 2 gives 4 = 0/0 which is not correct  

 

So there is only 1 pair of positive integers that satisfy the equation.