If the system of linear equations x + 2y + 3z = 4 2x + 3y + 4z = 5 3x + 4y + pz = q has infinitely many solutions, then the value of p + q is equal to

Step-by-step solution

For the system of equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero. = vmatrix 1 & 2 & 3 2 & 3 & 4 3 & 4 & p vmatrix = 0 Expanding along the first row: 1(3p - 16) - 2(2p - 12) + 3(8 - 9) = 0 3p - 16 - 4p + 24 - 3 = 0 -p + 5 = 0 p = 5 For infinitely many solutions, the system must be consistent. This means the third equation must be a linear combination of the first two equations. Let R₃ = aR₁ + bR₂ . Comparing the coefficients of x and y : a + 2b = 3 2a + 3b = 4 Sol