Linear equation system solver
Solve a system of linear equations with Gaussian elimination and see the rank, determinant and full solution.
Linear equation system solver
Paste a system of linear equations, one per line, and this tool builds the matrix, runs Gaussian elimination and prints the solution. It works equally well as an augmented-matrix solver: switch "Input as" to matrix mode and type rows of coefficients with the constant as the last value on each line, and skip writing out variable names entirely.
Every equation is written with the variables on the left and a plain number on the right, like "2x + 3y = 5" — any letters work as variable names, including things like x1 and x2, and a lone number on the left (such as "2x + 3 = 5") is moved across the equals sign automatically. Toggle "Output as" between exact fractions and fixed decimals, and set how many decimal places to show. Turn on "Show elimination steps" to see the augmented matrix after every pivot, and "Show rank and determinant" to see the rank of the coefficient matrix, the rank of the augmented matrix, and — for a square system — the determinant.
Not every system has exactly one answer. When the equations contradict each other the tool reports that plainly, along with the two ranks that prove it. When there are more unknowns than independent equations, it reports infinitely many solutions and, with "Parametric solution" turned on, writes the pivot variables in terms of the free ones so you can see the whole solution set rather than just one point in it. Systems are capped at 50 equations and 50 unknowns to keep the page responsive; larger systems should be split into smaller blocks.
Everything runs in your browser using exact fraction arithmetic, so results never drift the way floating-point elimination can on an ill-conditioned system. Nothing you type is uploaded anywhere. Copy the result, download it as a .txt file, or send the output back into the input to keep refining the system.