If the system of equations: x+y+z=5 x+2y+3z=9 x+3y+ z= has infinitely many solutions, then the value of + is:

Step-by-step solution

For the system to have infinitely many solutions, = 0 and _x = _y = _z = 0 . = vmatrix 1 & 1 & 1 1 & 2 & 3 1 & 3 & vmatrix = 1(2 - 9) - 1( - 3) + 1(3 - 2) = 2 - 9 - + 3 + 1 = - 5 Setting = 0 gives = 5 . Now, calculating _z : _z = vmatrix 1 & 1 & 5 1 & 2 & 9 1 & 3 & vmatrix = 1(2 - 27) - 1( - 9) + 5(3 - 2) _z = 2 - 27 - + 9 + 5 = - 13 Setting _z = 0 gives = 13 . For = 5 and = 13 , we can verify that _x = 0 and _y = 0 as well. Therefore, + = 5 + 13 = 18 . Answer: 18