If x, y and z are real numbers that satisfy the three equations array l (x)+ (y)+ (z)=6-( (x)+ (y)+ (z)) ^2(x)+ ^2(y)+ ^2(z)=6- ( ^2(x)+ ^2(y)+ ^2(z) ) ^3(x)+ ^3(y)+ ^3(z)=6- ( ^3(x)+ ^3(y)+ ^3(z) ) array . Find the value of the expression ( (x) (y) + (y) (z) + (z) (x) +3 (x) (y) (z) ) .

Step-by-step solution

Let x=a, y=b, z=c Given system of equation is equal to aligned a+b+c & =6- 1 a - 1 b - 1 c a^2+b^2+c^2 & =6- 1 a^2 - 1 b^2 - 1 c^2 a^3+b^3+c^3 & =6- 1 a^3 - 1 b^3 - 1 c^3 aligned From second equation we complete squares to get (a- 1 a )^2+ (b- 1 b )^2+ (c- 1 c )^2=0 a=b=c= 1 Now rearranging 3^ rd equation and adding 3 time first equation, we get aligned a^3+b^3+c^3+ 1 a^2 + 1 b^2 + 1 c^2 +3 (a+ 1 a +b+ 1 b +c+ 1 c ) & =6+18 (a+ 1 a )^3+ (b+ 1 b )^3+ (c+ 1 c )^3 & =24 aligned Therefore a=b=c=1 Hence, [ (x) (y) + (y)