System of Equations Calculator

A system of two linear equations asks for values of two variables that satisfy both equations at the same time. In standard form, each equation combines a coefficient of x, a coefficient of y, and a constant on the other side. Geometrically, each equation represents a line. A unique solution is the point where two nonparallel lines intersect. This calculator uses Cramer's rule for a two-by-two system. It first computes the coefficient determinant by multiplying the first x coefficient by the second y coefficient and subtracting the product of the second x coefficient and first y coefficient. When that determinant is nonzero, replacing one coefficient column at a time with the constants gives two additional determinants. Dividing those by the coefficient determinant yields x and y. The displayed answer is rounded to six decimal places with unnecessary trailing zeroes removed. A zero determinant means the coefficient rows are linearly dependent, so there is no unique intersection. The two equations may describe the same line, giving infinitely many solutions, or distinct parallel lines, giving no solution. Phase 1 reports both cases as no unique solution because either case cannot produce one ordered pair. Substitution into the original equations can distinguish them, but the central warning remains the same: division by the coefficient determinant is not valid. Systems of equations model situations where several constraints apply simultaneously. Typical examples include finding ticket prices from total attendance and revenue, separating mixtures, balancing supply and demand, comparing phone plans, locating intersections, and solving elementary circuit equations. Larger systems appear in engineering, economics, computer graphics, statistics, and numerical simulation. The two-variable case is especially useful for learning elimination, substitution, determinants, and matrix notation because every step can be checked by hand. Enter decimals, integers, positive values, negative values, or zero as needed. Be careful to move every term into the stated standard form before entering coefficients. If an equation lacks one variable, its coefficient is zero rather than blank. Signs must follow the rearranged equation exactly. After calculating, verify a unique result by substituting x and y back into both original equations. Small floating-point rounding can appear for fractional solutions, but six displayed decimal places are sufficient for typical educational and practical problems.