If the system of linear equations: x+y+z=6 , x+2y+5z=10 , 2x+3y+ z= has infinitely many solutions, then the value of + equals:

Step-by-step solution

For the system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and _x = _y = _z = 0 . = vmatrix 1 & 1 & 1 1 & 2 & 5 2 & 3 & vmatrix = 0 Expanding the determinant: 1(2 - 15) - 1( - 10) + 1(3 - 4) = 0 - 6 = 0 = 6 Now, setting _z = 0 : _z = vmatrix 1 & 1 & 6 1 & 2 & 10 2 & 3 & vmatrix = 0 Expanding the determinant: 1(2 - 30) - 1( - 20) + 6(3 - 4) = 0 - 16 = 0 = 16 Therefore, + = 6 + 16 = 22 . Answer: 22