Elimination Method Calculator

The elimination method solves a system of simultaneous linear equations by removing one variable. Each equation represents a straight line, and a solution is a point that satisfies both equations. Multiplying an entire equation by a nonzero number does not change its line, so the equations can be scaled until one pair of coefficients are opposites. Adding the scaled equations then eliminates that variable and leaves a one-variable equation. Substitution into either original equation produces the remaining coordinate. This calculator accepts two equations in standard form, with an x coefficient, a y coefficient, and a constant for each row. It uses determinant arithmetic, which is algebraically equivalent to elimination for a two-equation system. The main determinant compares the coefficient rows. When it is nonzero, the two lines have different slopes and meet at exactly one point. The calculator divides two related determinants by the main determinant to obtain x and y, retaining full floating-point precision before rounding the displayed values. A zero determinant requires additional interpretation. If the coefficient rows and constants are proportional, both equations describe the same line and every point on that line satisfies the system. The result is infinitely many solutions. If the coefficient rows are proportional but their constants are not, the lines are parallel and distinct, so no ordered pair can satisfy both equations. Reporting these cases is important because dividing by a zero determinant would otherwise produce a misleading numerical result. Elimination is useful in algebra classes, coordinate geometry, circuit analysis, mixture problems, pricing models, and any situation with two unknown quantities constrained by two linear relationships. Negative, decimal, and zero coefficients are valid. To check a unique answer, substitute the reported x and y values into both original equations and confirm that each left side equals its constant, allowing for small rounding differences. For hand calculations, choose the variable whose coefficients are easiest to match, scale carefully, and apply every multiplication to all terms in that equation. This tool provides a quick numerical answer while preserving the core classification of the system.