No, a system of linear equations cannot have exactly two solutions. In the context of linear algebra, the solutions to a system of linear equations can fall into three categories:
- Exactly one solution: This occurs when the equations represent lines that intersect at a single point. These lines are neither parallel nor identical.
- No solutions: This is the case when the equations represent parallel lines that never intersect, indicating that there is no common solution.
- Infinitely many solutions: This scenario arises when the equations represent the same line, implying that there are countless points that satisfy the equations.
In summary, while a system can have unique, none, or infinitely many solutions, it cannot have exactly two solutions due to the properties of linear equations in a plane, where the relationships between the equations adhere to the concepts of intersection and parallelism.