How many solutions can a system of linear equations have?

A system of linear equations can have three possible types of solutions:

One unique solution: This occurs when the system of equations is consistent (has at least one solution) and the equations represent different lines (or hyperplanes in higher dimensions) that intersect at exactly one point. This happens when the determinant of the coefficient matrix is non-zero. In other words, the system is independent and has exactly one solution.

Infinitely many solutions: This occurs when the system is consistent but the equations represent the same line or plane (in higher dimensions). This can happen when the equations are dependent, meaning that one equation is a linear combination of the others. In such cases, there are infinitely many solutions because the variables can take on an infinite number of values along the line or plane.

No solution: This occurs when the system is inconsistent. The equations represent parallel lines (or hyperplanes in higher dimensions) that never intersect. In this case, there is no point that satisfies all the equations simultaneously.

Thus, the possible number of solutions for a system of linear equations is:

One unique solution
Infinitely many solutions
No solution
The specific number of solutions depends on the properties of the system, such as the rank of the coefficient matrix and the consistency of the equations.