Question

Can a linear system of three equations have exactly two solutions? Explain why or why not.

Answer 1

Answer: A system of linear equations can't have exactly two solutions. A linear equation with two variables has an infinite number of solutions. However, systems of two linear equations with two variables can have a single solution that satisfies both equations.

Answer 2

No, because it would be inconsistent with the geometric interpretation of the system.

Explanation:

A linear system of three equations cannot have exactly two solutions because it would be inconsistent with the geometric interpretation of the system.

Geometrically, a system of three linear equations can be represented by three planes in three-dimensional space.

If the system has two solutions, then the three planes would intersect in a line.

However, this is impossible because three planes can only intersect in a point or not at all.

Therefore, a linear system of three equations cannot have exactly two solutions.