By eliminating a,b,c from the homogenous equations x=ab−c,y=bc−a,z=ca−b  where  a,b,c  not all zero

x=ab−c⇒a=xb−xc⇒a−xb+xc=0→(1)y=bc−a⇒b=yc−ya⇒ya+b−yc=0→(2)z=ca−b⇒c=za−zb⇒za−zb−c=0→(3) By eliminating a.b,c from (1), (2), (3), we get 1−xxy1−yz−z−1=0⇒1(−1−yz)+x(−y+yz)+x(−yz−z)=0⇒−1−yz−xy+xyz−xyz−zx=0⇒xy+yz+zx=−1