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Why can a system of two linear equations have infinitely many solutions?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — algebra This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In algebra, a system of two linear equations has infinitely many solutions when the two equations represent the exact same line.1

Geometrically, a system of two linear equations in two variables corresponds to the intersection of two lines. When these lines are distinct and not parallel, they intersect at a single point, providing a unique solution. However, if the equations are multiples of one another, they describe the same set of points in the coordinate plane. In this scenario, every point on the line satisfies both equations simultaneously, resulting in an infinite number of solutions.1

Algebraically, this occurs when the determinant of the system is equal to zero. When the determinant is zero, the system is either inconsistent (if the lines are parallel and distinct, resulting in no solution) or dependent (if the lines are identical, resulting in infinitely many solutions).1