<p>The system of equations<br>\(x + 8y + 7z = 0\)<br>\(9x + 2y + 3z = 0\)<br>\(x + y + z = 0\)<br>has a non-trivial solution. If \(x = a, y = b, z = c\) is the solution and \((a, b, c)\) lies on the plane \(x + 2y + z = 6\), find the value of \(2a + b + c\).</p>
Step-by-Step Solution
Key Concept: A homogeneous system has non-trivial solutions only when the coefficient matrix determinant is zero. Use this constraint along with the plane equation to uniquely determine the ratio a:b:c, then apply the plane condition to find the scalar multiple.
<p><strong>Step 1:</strong> For non-trivial solution, the coefficient matrix determinant must be zero:</p><p>$$\begin{vmatrix} 1 & 8 & 7 \\ 9 & 2 & 3 \\ 1 & 1 & 1 \end{vmatrix} = 1(2-3) - 8(9-3) + 7(9-2) = -1 - 48 + 49 = 0$$ ✓</p><p><strong>Step 2:</strong> Find the null space. From equation (3): x + y + z = 0, so z = -x - y. Substitute into equation (1):</p><p>x + 8y + 7(-x-y) = 0 → x + 8y - 7x - 7y = 0 → -6x + y = 0 → y = 6x</p><p>Therefore: z = -x - 6x = -7x</p><p>General solution: (x, y, z) = t(1, 6, -7) where t ∈ ℝ</p><p><strong>Step 3:</strong> Given (a, b, c) = (t, 6t, -7t) lies on plane x + 2y + z = 6:</p><p>t + 2(6t) + (-7t) = 6 → t + 12t - 7t = 6 → 6t = 6 → t = 1</p><p><strong>Step 4:</strong> Therefore a = 1, b = 6, c = -7</p><p>$$2a + b + c = 2(1) + 6 + (-7) = 1$$</p>
Correct Answer: 1