The number of triplets (α, β, γ) satisfying the following constraints<br>2α - β + 3γ = 4<br>α + β - 3γ = -1<br>5α - β + 3γ = 7<br>αβγ ≤ 0<br>& α, β, γ ∈ I
Step-by-Step Solution
Key Concept: Solve the system of linear equations to find unique values for \alpha, \beta, and \gamma, then check the inequality constraint.
Adding the first two equations: (2\alpha - \beta + 3\gamma) + (\alpha + \beta - 3\gamma) = 4 - 1 => 3\alpha = 3 => \alpha = 1. Substituting \alpha = 1 into the first two equations: 2 - \beta + 3\gamma = 4 => -\beta + 3\gamma = 2 and 1 + \beta - 3\gamma = -1 => \beta - 3\gamma = -2. These are consistent. Using the third equation: 5(1) - \beta + 3\gamma = 7 => 5 - \beta + 3\gamma = 7 => -\beta + 3\gamma = 2. All equations reduce to \beta - 3\gamma = -2, or \beta = 3\gamma - 2. Since \alpha = 1, the condition \alpha\beta\gamma \leq 0 becomes 1(3\gamma - 2)\gamma \leq 0, which is \gamma(3\gamma - 2) \leq 0. This inequality holds for 0 \leq \gamma \leq 2/3. Since \gamma must be an integer, the only possible value is \gamma = 0. If \gamma = 0, then \beta = 3(0) - 2 = -2. Thus, the only triplet is (1, -2, 0). The number of such triplets is 1.
Correct Answer: 1