Calculator Tools
System of Equations Solver
Solve 2x2 or 3x3 systems of linear equations step by step. Exact fractions, classified outcomes, and a full elimination trace.
Quick examples
System size
Input mode
Enter 2 equations in x, y
Write equations like 2x + 3y = 7. Coefficients can be integers, decimals, or fractions (1/2). Implicit coefficients (x means 1x) are fine. Terms on either side of the equals sign are accepted.
Parsed as: 2x + 3 y = 7
Parsed as: 1x - 1 y = 1
- x
- 2
- y
- 1
Step-by-step elimination
Each step transforms the augmented matrix toward reduced row-echelon form using exact rational arithmetic.
Step 1
Initial augmented matrix
x y | rhs 2 3 7 1 -1 1 Step 2
Divide row 1 by 2 so the pivot in column 1 becomes 1
x y | rhs 1 3/2 7/2 1 -1 1 Step 3
Eliminate column 1 from every other row using row 1
x y | rhs 1 3/2 7/2 0 -5/2 -5/2 Step 4
Divide row 2 by -5/2 so the pivot in column 2 becomes 1
x y | rhs 1 3/2 7/2 0 1 1 Step 5
Eliminate column 2 from every other row using row 2
x y | rhs 1 0 2 0 1 1
Plain-text summary
Copy the parsed equations and the solution as a single block of text. Useful for pasting into homework, notes, or a chat.
System of Equations Solver
Input
(1) 2x + 3 y = 7
(2) 1x - 1 y = 1
Result
Unique solution
x = 2 (2)
y = 1 (1)
Computed in your browser. Inputs are not uploaded.Notes and caveats
- Exact answers. Coefficients are stored as fractions (BigInt numerator and denominator), so the solver does not lose digits to floating-point rounding. Decimals are converted to exact fractions on entry.
- Three textbook outcomes. A consistent system with full rank has a unique solution; a zero-row with a non-zero right-hand side is inconsistent; and a consistent system with rank below the number of variables has infinitely many solutions, parameterised by free variables.
- Allowed coefficients. Integers (7), decimals (-0.25), and fractions (3/4) are accepted. Mixed numbers are not parsed; write them as improper fractions.
- Local only. Nothing you type is uploaded. The matrix is solved entirely in your browser.
How to use
- Choose the system size: 2 equations in x, y or 3 equations in x, y, z.
- Pick an input mode. Equations mode accepts free-form lines like '2x + 3y = 7' and '1/2 x - y = 0'. Augmented-matrix mode lets you fill the coefficient grid directly.
- Try a preset chip ('Two equations, unique', 'Three, no solution', 'Three, infinitely many') to load a worked example.
- Watch the per-row echo under each input to confirm how the parser read your line.
- Read the solution card: unique answers list every variable as an exact fraction and a decimal; no-solution and infinitely-many cases are explained with a parameterised form.
- Open the step-by-step elimination panel to see every row operation that took the matrix to reduced row-echelon form.
- Click Copy summary to copy a plain-text version of the input, the solution, and the parameterisation into your clipboard.
About this tool
System of Equations Solver solves 2-variable and 3-variable linear systems entered either as equations (like 2x + 3y = 7) or directly as an augmented matrix. Coefficients can be integers, decimals, or fractions; every value is converted to an exact rational (BigInt numerator and denominator) before solving, so the answers are not polluted by floating-point rounding. The solver runs Gaussian elimination with partial pivoting and reduces the augmented matrix to reduced row-echelon form. Each row operation is recorded so the tool can show every step: swaps, scaling a pivot to 1, and elimination passes that clear a column. After reduction, the rank test classifies the system into one of the three textbook outcomes: a unique solution when the coefficient matrix and the augmented matrix have full rank equal to the number of variables; no solution when a zero-row carries a non-zero right-hand side (the equations contradict each other); or infinitely many solutions when the rank is below the number of variables, in which case the tool names free parameters (t1, t2, ...) and expresses each basic variable as a clean linear function of those parameters. The parser is tolerant: implicit coefficients work (x means 1x), terms on either side of the equals sign are accepted and rearranged, decimals and fractions can mix in the same equation, and a per-row echo shows how each line was interpreted so typos surface immediately. The Copy summary button packages the parsed equations, the solution (or the parameterized general solution), and the matrix form into a single plain-text block suitable for pasting into homework, notes, or a study chat. Useful for algebra and pre-calc students, for engineers checking a quick balance, for anyone working through linear-algebra examples by hand, and for confirming that a textbook answer key matches an exact rational result. All computation happens locally in your browser; the equations you type are never uploaded, logged, or sent to a server.
Free to use. Works in your browser. No signup, no login.
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