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An overdetermined system is a set of equations that has more equations than unknown variables. In linear algebra, this typically refers to a system of linear equations where the number of equations exceeds the number of unknowns. Such systems may have a unique solution, no solution, or infinitely many solutions, but generally, they are more likely to have no solution than a system with an equal number of equations and unknowns.

When an overdetermined system has a solution, it means that all equations are consistent with each other. However, if the equations are inconsistent, there is no single set of values for the unknowns that satisfies all equations simultaneously. In practice, real-world data often leads to overdetermined systems due to measurement errors or noise. In these cases, finding an exact solution is impossible.

Instead of seeking an exact solution, the focus shifts to finding a "best approximate solution." This is commonly achieved using methods like the method of least squares. The least squares method finds the values of the unknowns that minimize the sum of the squares of the differences between the observed values and the values predicted by the equations. This approach provides a compromise that best fits the data, even if it doesn't perfectly satisfy every equation. Overdetermined systems are prevalent in fields such as statistics, engineering, and machine learning, where numerous measurements are used to estimate a smaller number of parameters.