No, a system of linear equations cannot have exactly two solutions.
Here’s why: A system of linear equations can have three possible outcomes: no solution, exactly one solution, or infinitely many solutions. These outcomes are based on the relationship between the lines represented by the equations.
1. **No Solution:** This occurs when the lines are parallel and never intersect. Each equation describes a different line with the same slope, which means they will never meet.
2. **Exactly One Solution:** This happens when the lines intersect at a single point. In this case, the equations represent lines with different slopes.
3. **Infinitely Many Solutions:** This situation arises when the lines overlap completely, meaning they are essentially the same line represented in different forms.
Since these are the only scenarios possible, having exactly two solutions falls outside the basic principles of linear equations. If two solutions existed, they would imply multiple intersections, which contradicts the linear nature of the equations involved.