Systems of linear equations
Write the coefficients of a 2x2 or 3x3 system and get the solution in exact fractions, the steps of the method you pick and the check. Systems with no solution or with infinitely many are explained too.
Fields accept whole numbers, decimals with a dot and fractions such as 3/2. An empty field counts as 0, and the result always rewrites the system as it was read.
How to use it
Pick whether the system has 2 or 3 equations, then write the coefficients in the grid: each row is one equation and the last field after the equals sign is the constant term. Whole numbers, decimals and fractions written as 3/2 all work. If an unknown does not appear in an equation, leave the field empty or write 0. The result box always rewrites the system as it was read, so a wrong sign shows up before you even look at the solution.
Why the solution comes as fractions
Take the system 0.1x + 0.2y = 0.3 with x - y = 0: the solution is x = 1, yet a calculator working in floating point answers 0.9999999999999998, because in binary 0.1 + 0.2 is not exactly 0.3. Here the arithmetic runs on fractions with whole numerator and denominator, so that solution comes out as exactly 1 and 1/3 stays 1/3 rather than turning into 0.333. Next to the fraction there is the decimal value, introduced by = when the decimal writing really ends and by ≈ when it is rounded, so nothing rounded is passed off as exact.
The three methods, and when each one pays off
Substitution takes one unknown out of an equation and puts it into the others, and it feels natural when some coefficient is already 1. Elimination works on the rows of the coefficient table by adding them to each other (this is Gaussian elimination, the same thing school calls the addition method) and it scales nicely to the 3x3 case. Cramer rule computes one determinant for the system and one per unknown, which is handy when you only need a single unknown, but it works only if the determinant D is not zero. The answer never depends on the method, only the story of the steps does.
No solution and infinitely many, without the wrong shortcut
A system has no solution when the equations contradict each other (two distinct parallel lines) and infinitely many when one equation just repeats another (the same line), in which case the solutions are written with a parameter. Many textbooks say that D, Dx and Dy all zero means infinitely many solutions, which is not always true: the system 0x + 0y = 0 and 0x + 0y = 1 has all determinants equal to zero and no solution at all. That is why the case is decided here by actually reducing the equations and counting how many independent ones are left, while the determinant is reported as a note.
Stated limits
This tool handles square linear systems, that is 2 equations with 2 unknowns or 3 with 3 unknowns. It does not handle powers or products of unknowns (no x², no xy), systems with more equations than unknowns, inequalities or letter parameters inside the coefficients. Each number may carry at most 12 integer digits and 10 decimals. Everything is computed inside your browser and no data is sent anywhere. Redoing the check by hand stays a good habit, and here it is already written out step by step.