Understanding the System of Equations
The given equations form a system of linear equations. Let's analyze them step by step.
Equations Provided:
- 3x + 4y - z - 6w = 0
- 2x + 3y + 2z - 3w = 0
- 2x + y - 14z - 9w = 0
- x + y + 13z + 3w = 0
Objective:
To determine the solution set for the variables x, y, z, and w.
Steps to Solve:
- Formulate the Augmented Matrix:
Convert the system of equations into an augmented matrix for easier manipulation.
- Row Reduction:
Use Gaussian elimination or row reduction to simplify the matrix to Row Echelon Form (REF) or Reduced Row Echelon Form (RREF).
- Identify Solutions:
Analyze the resulting matrix to identify if the system has a unique solution, infinite solutions, or no solution at all.
Important Considerations:
- Rank of the Matrix:
The rank of the coefficient matrix and the augmented matrix can indicate the nature of the solutions.
- Dependent and Independent Variables:
If the system has infinite solutions, some variables can be expressed in terms of others, indicating dependencies.
Conclusion:
The systematic approach to solving this set of equations will reveal the relationships between the variables and provide insight into the solution space. By following the outlined steps, one can efficiently tackle similar linear equation problems in UPSC and other competitive exams.