If a , b , c are non-zero real numbers and if the equations (a-1) x=y+z,(b-1) y=z+x,(c-1) z=x+y have a non-trivial solution, then a b+b c+c a=

Step-by-step solution

We have system of equation, (a-1) x-y-z=0 ; x-(b-1) y+z=0 and x+y-(c-1) z=0 It is a homogeneous system of equations. Now, for non-trivial solution. aligned & | array ccc a-1 & -1 & -1 1 & -(b-1) & 1 1 & 1 & -(c-1) array |=0 & | array ccc a-1 & -1 & -1 1 & 1-b & 1 1 & 1 & 1-c array |=0 & (a-1)[(1-b)(1-c)-1]+1[1-c-1] & -[1-(1-b)]=0 & (a-1)(1-b)(1-c)-a+1-c-b=0 & (a-1)(1-c-b+b c)-a-b-c+1=0 & a-a c-a b+a b c-1+c+b-b c-a-b-c+1=0 & a b+b c+c a=a b c aligned