Question

Consider the system of equations:
\[ x + 2y - z = 3 \\ 2x + 4y - 2z = 7 \\ 3x + 6y - 3z = 9 \]
Which of the following statements is true about the system?

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

Hint:

When you have a system of equations that are multiples of each other, the system has infinitely many solutions unless there is an inconsistency.

Answer 1

The Correct Option is B

We are given the system of equations: \[ x + 2y - z = 3 \quad \cdots (1) \\ 2x + 4y - 2z = 7 \quad \cdots (2) \\ 3x + 6y - 3z = 9 \quad \cdots (3) \] Looking at equations (2) and (3), notice that they are scalar multiples of equation (1). Specifically: \[ 2x + 4y - 2z = 7 \text{ is } 2 \times (x + 2y - z = 3) \] and similarly, \[ 3x + 6y - 3z = 9 \text{ is } 3 \times (x + 2y - z = 3) \] Thus, the system is not independent and represents a set of equations that are linearly dependent. Since the system has fewer independent equations than unknowns, it has infinitely many solutions.