Question

Consider the system of equations: \[ \begin{aligned} x + y + z &= 6 \\ 2x + 2y + 3z &= 13 \\ 3x + 4y + 5z &= 20 \end{aligned} \] Which of the following statements is true about the solution to this system?

• A. The system has a unique solution.
• B. The system has no solution.
• C. The system has infinitely many solutions.
• D. The system has finitely many solutions but not unique.

Hint:

A square system with full rank has a unique solution.

Answer 1

The Correct Option is A

This is a system of three linear equations in three variables. Writing it in matrix form and solving using Gaussian elimination or checking the determinant of the coefficient matrix shows that the system is consistent and the coefficient matrix is non-singular (i.e., has full rank). Therefore, the system has a unique solution.