If the system of equations x + y - 3 = 0, (1 + K)x + (2 + K)y - 8 = 0 & x - (1 + K)y + (2 + K) = 0 is consistent then the value of K may be -
Step-by-Step Solution
Key Concept: For a system of linear equations to be consistent, the determinant of the coefficient matrix must be zero if the system is homogeneous or if we are looking for a non-trivial solution, or more generally, the augmented matrix must have the same rank as the coefficient matrix. For three lines to be consistent, they must either be concurrent or parallel.
The system of equations is: x + y = 3, (1+K)x + (2+K)y = 8, x - (1+K)y = -(2+K). For the system to be consistent, the determinant of the coefficient matrix must be zero: |1 1 -3; (1+K) (2+K) -8; 1 -(1+K) (2+K)| = 0. Solving this determinant gives the values of K for which the system is consistent. Evaluating the determinant: 1((2+K)(2+K) - 8(-(1+K))) - 1((1+K)(2+K) - (-8)) - 3((1+K)(-(1+K)) - (2+K)) = 0. Simplifying this leads to a quadratic equation in K. Solving for K gives K=1 as one of the possible values.
Correct Answer: A