In the following system of equations x,y,z are variables and k is a constant. All values are real numbers.
- x + y*z = k
- y + x*z = k
- z + x*y = k
Determine the relationship between k and the number of the solutions of the system.
In general the 5 solutions for any k are:
(1,1,k-1), (1,k-1,1) and (k-1,1,1) and (x,x,x)
where x = -1/2 +/- sqrt(1+4k)/2
exceptions:
When k < -1/4, there are only 3 real solutions:
(1,1,k-1), (1,k-1,1) and (k-1,1,1)
When k = -1/4, there are four solutions:
(1,1,-5/4), (1,-5/4,1), (-5/4,1,1) and (-1/2,-1/2,-1/2)
when k = 2 there are only three solutions
(1,1,1), (x,x,x) and (y,y,y)
where x = (-1+sqrt(5))/2
and y = (-1-sqrt(5))/2
Edited on October 26, 2009, 3:26 pm
Spoiler (no proof)