Given positive real numbers x, y, and z that satisfy the following system of equations:x² + xy + y² = 5,y² + yz + z² = 4,z² + zx + x² = 1,Find x + y + z?

• square root 10 will be answer.
  sum all the expression
  it will form like (x+y+z)^2=10
  so x+y+z will be square root 10
• 9 years agoHelpfull: Yes(0) No(1)
• Please note that solution given by Kaushal Kishor is definitely wrong.
  If you add all three equations, you get 2x^2 + 2y^2 + 2z^2 + xy + yz + zx = 10.
  On dividing both sides, x^2 + y^2 + z^2 + xy/2 + yz/2 + zx/2 = 5
  Now, pleas note: (x+y+z)^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx
  So, we can write the above equation as (x+y+z)^2 - 3/2*(xy + yz + zx) = 5
  or (x+y+z)^2 = 5 + 3/2*(xy + yz + zx)
  
  Now, my computer algebra system gives the following answers:
  w is defined as sum of x, y and z. Or w = x+y+z
  
  Solution 1
  x = 0.7406217908365105 + [-2.335e-012,+2.335e-012]
  y = 1.771793098836831 + [-1.358e-012,+1.358e-012]
  z = 0.3968980215030719 + [-2.555e-013,+2.555e-013]
  w = 2.909312911176413 + [-2.549e-013,+2.549e-013]
  
  precision: 4.67e-012, elapsed time: 3 ms
  
  Solution 2
  x = -0.1248275892608901 + [-3.622e-015,+3.622e-015]
  y = 2.29586707480433 + [-1.332e-015,+1.332e-015]
  z = -0.9317258106159639 + [-1.665e-015,+1.665e-015]
  w = 1.239313674927476 + [-1.554e-015,+1.776e-015]
  
  precision: 7.24e-015, elapsed time: 7 ms
  
  Solution 3
  x = 0.1248275892608903 + [-2.304e-015,+2.318e-015]
  y = -2.29586707480433 + [-4.441e-016,+4.441e-016]
  z = 0.9317258106159638 + [-7.772e-016,+7.772e-016]
  w = -1.239313674927476 + [-6.661e-016,+6.661e-016]
  
  precision: 4.62e-015, elapsed time: 13 ms
  
  Solution 4
  x = -0.7406217908365104 + [-2.324e-012,+2.324e-012]
  y = -1.771793098836831 + [-1.351e-012,+1.351e-012]
  z = -0.3968980215030719 + [-2.444e-013,+2.444e-013]
  w = -2.909312911176413 + [-2.451e-013,+2.456e-013]
  
  precision: 4.65e-012, elapsed time: 17 ms
  
• 9 years agoHelpfull: Yes(0) No(0)