Simultaneous equations are a set of equations containing multiple variables, where all equations must be solved together to find values that satisfy every equation at the same time. For example, the system 2x + 3y = 12 and x − y = 1 has a unique solution (x = 3, y = 2) that makes both equations true simultaneously. They are also called a system of linear equations when all variables appear to the first power only.

There are three main methods: (1) Substitution — rearrange one equation to make a variable the subject, substitute into the other equation, solve, then back-substitute. (2) Elimination — multiply equations to make coefficients match, then add or subtract to eliminate a variable. (3) Cramer’s Rule (determinants) — use matrix determinants to find each variable directly: x = Dx/D, y = Dy/D. This calculator uses Cramer’s rule as the primary method and shows step-by-step working for all methods.

Simultaneous equations have no solution when the lines represented by the equations are parallel — meaning they have the same gradient but different intercepts. Mathematically, this occurs when the determinant D = 0 but Dx or Dy is non-zero. For example, 2x + 3y = 6 and 4x + 6y = 12 represent the same line (infinite solutions), while 2x + 3y = 6 and 4x + 6y = 15 are parallel with no solution. The calculator will identify these cases automatically.

Yes. This calculator supports both 2×2 systems (two equations with two variables x and y) and 3×3 systems (three equations with three variables x, y, and z). For 3×3 systems, it uses the 3×3 determinant method (expansion by minors) to solve using Cramer’s rule. Simply select ‘3 variables’ from the dropdown and enter the coefficients for all three equations.

Cramer’s rule is a method for solving systems of linear equations using determinants. For a 2×2 system ax + by = e, cx + dy = f: the main determinant D = ad − bc, Dx = ed − bf, Dy = af − ec, then x = Dx/D and y = Dy/D. It extends to 3×3 systems using 3×3 determinants. Cramer’s rule only works when D ≠ 0; if D = 0, the system has either no solution or infinitely many solutions. This calculator uses Cramer’s rule as its primary solving method.

Simply enter negative numbers directly into the coefficient fields. For example, for the equation 3x − 2y = 7, enter 3 for the x coefficient, −2 for the y coefficient, and 7 for the constant. The calculator accepts both positive and negative integers and decimals. You can also enter equations in rearranged forms — just ensure all variable terms are on one side and the constant on the other.

You can enter decimal equivalents of fractions directly. For example, ½x + ⅓y = 5 can be entered as 0.5 for the x coefficient and 0.3333 for the y coefficient. For better accuracy, multiply all equations through by a common denominator first to convert fractions to integers — for example, multiply by 6 to get 3x + 2y = 30, which gives exact results.

Substitute the calculated values of x and y (and z for 3-variable systems) back into each original equation. If all equations evaluate correctly, the solution is verified. This calculator includes an automatic verification step that shows each equation with the solution substituted in, confirming that both sides are equal. This is the standard mathematical check for any simultaneous equations solution.