Find: x^2+y^2+z^2 If x^2+xy+xz=135 y^2+yz+yx=351 z^2+zx+zy=243

If we add all the three equations:

Approved

Adding The Three Equations That We Are Given x^2+xy+xz+y^2+yz+yx+z^2+zx+zy = 135+351+243 x^2+y^2+z^2+2(xy+yz+xz) = 729 But This Is The Identity For (x+y+z)^2 (x+y+z)^2 = 729 Taking Square Root x+y+z = 27 Now In The First Equation Take x common x(x+y+z) = 135 Put Value Of x+y+z 27x = 135 x = 5 Now Take y common is second equation y(x+y+z) = 351 27y = 351 y = 13 Similarly Take z common is third z(x+y+z) = 243 27z = 243 z = 9 Now x^2+Y^2+z^2 = 25+169+81 = 275

Approved