System of linear equations with two unknowns

A system of two linear equations with two unknowns is a pair of first-degree equations that share the same two variables, usually written as x and y, which we try to satisfy simultaneously. Because each equation is linear, it represents a straight line when drawn in the coordinate plane, so solving the system means finding the point where those two lines meet. Most systems have exactly one solution, but there are two special cases: if the lines are parallel there is no solution, and if the two equations describe the very same line there are infinitely many. Whether a unique solution exists depends only on the coefficients, specifically on whether one equation is a simple multiple of the other. Common ways to solve such a system by hand include substitution, elimination (adding or subtracting the equations), and Cramer's rule using determinants. These systems appear constantly in practice, from mixing two ingredients in the right proportions to balancing supply and demand or splitting a total between two quantities. They also form the foundation of linear algebra, where the same ideas extend to many equations and many unknowns solved with matrices.