Why can a system of two linear equations have no solution?

Created from Britannica articles using AI. AI can make mistakes, so verify using Britannica articles.

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 is considered inconsistent if it has no solution. Geometrically, these equations represent lines in a two-dimensional plane; a system has no solution when these lines are parallel and distinct, meaning they never intersect.12

Algebraically, this occurs when the determinant of the system—the denominator used in solving for the variables—is equal to zero. When the determinant is zero, the system either has no solution or infinitely many solutions (if the equations represent the same line). In the context of Cramer’s rule, a zero determinant indicates that there is no unique set of values that satisfies all equations in the system simultaneously.32