System of Equations Solver

A system of equations is a set of two or more equations that must all be satisfied at the same time, and solving it means finding the values of the unknowns that make every equation true simultaneously. Two-variable linear systems of the form ax plus by equals c arise constantly in maths courses and practical problems: finding where two lines intersect on a graph, working backwards from combined totals to find individual prices, solving mixture problems, balancing supply and demand, and many engineering and physics problems. This calculator solves a system of two linear equations in two variables using Cramer's rule, a determinant-based method that expresses each unknown directly as a ratio of two values computed from the coefficients. You enter the coefficient of x, the coefficient of y, and the constant on the right-hand side for each of the two equations, and the calculator returns the values of x and y, the main determinant of the coefficient matrix, and the step-by-step working so you can follow exactly how the answer was reached. Negative coefficients work fine; simply enter a negative number. If the main determinant is zero, the equations either contradict each other or are equivalent to the same line, and no unique solution exists. The calculator detects this and tells you rather than returning a meaningless result. Enter your coefficients to see the solution update as you type.

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Equation 1:   ax + by = c

Equation 2:   dx + ey = f

Working: D = (2)(−1) − (1)(3) = −5. Dx = (12)(−1) − (1)(3) = −15. Dy = (2)(1) − (1)(12) = −10. x = −15/−5 = 3. y = −10/−5 = 2.