Question
Let S be the set of all integer solutions, (x, y, z), of the system of equations x – 2y + 5z = 0 –2x + 4y + z = 0 –7x + 14y + 9z = 0 such that 15 x 2 + y 2 + z 2 150. Then, the number of elements in the set S is equal to ______ .
Answer: 2
Solution
Key Concept: Solving a system of homogeneous linear equations and finding integer solutions that satisfy a given inequality. A system of homogeneous linear equations has non-trivial solutions if and only if the determinant of the coefficient matrix is zero. The general solution represents a subspace, and we need to find integer points within this subspace that fall within a specified range.
1. Analyze the System of Homogeneous Linear Equations
We are given the following system of linear homogeneous equations:
**Explanation