If the system of equations <br> x + y + z = 6 <br> 2x + 5y + αz = β <br> x + 2y + 3z = 14 <br> has infinitely many solutions, then α + β is equal to :
Step-by-Step Solution
Key Concept: For a system of linear equations to have infinitely many solutions, the determinant of the coefficient matrix must be zero, and the augmented matrix must satisfy the condition for consistency (rank(A) = rank(A|B) < number of variables).
The system is: <br> x + y + z = 6 <br> 2x + 5y + \alpha z = \beta <br> x + 2y + 3z = 14 <br> For infinitely many solutions, the determinant of the coefficient matrix must be 0: <br> |1 1 1; 2 5 \alpha; 1 2 3| = 0 <br> 1(15 - 2\alpha) - 1(6 - \alpha) + 1(4 - 5) = 0 <br> 15 - 2\alpha - 6 + \alpha - 1 = 0 <br> 8 - \alpha = 0 => \alpha = 8 <br> Now, the system becomes: <br> x + y + z = 6 (i) <br> 2x + 5y + 8z = \beta (ii) <br> x + 2y + 3z = 14 (iii) <br> Subtract (i) from (iii): y + 2z = 8 => y = 8 - 2z <br> Substitute y into (i): x + 8 - 2z + z = 6 => x - z = -2 => x = z - 2 <br> Substitute x and y into (ii): 2(z - 2) + 5(8 - 2z) + 8z = \beta <br> 2z - 4 + 40 - 10z + 8z = \beta <br> 36 = \beta <br> Thus, \alpha = 8 and \beta = 36. <br> \alpha + \beta = 8 + 36 = 44.
Correct Answer: 3