If the system of linear equations x - 2y + 3z = 9 2x + y + z = b x - 7y + az = 24 has infinitely many solutions, then the value of a + b is

Step-by-step solution

For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero ( = 0 ), and _x = _y = _z = 0 . First, setting = 0 : = vmatrix 1 & -2 & 3 2 & 1 & 1 1 & -7 & a vmatrix = 0 Expanding along the first row: 1(a - (-7)) - (-2)(2a - 1) + 3(-14 - 1) = 0 a + 7 + 4a - 2 - 45 = 0 5a - 40 = 0 a = 8 Now, setting _z = 0 : _z = vmatrix 1 & -2 & 9 2 & 1 & b 1 & -7 & 24 vmatrix = 0 Expanding along the first row: 1(24 + 7b) - (-2)(48 - b) + 9(-14 - 1) = 0 24 + 7b + 96 - 2b -