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Linear Systems Solver (2×2 / 3×3)

Solve 2×2 or 3×3 linear systems with Gaussian elimination and review solution classification, parametric forms, residuals, and elimination steps in one place.

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Educational use only. Double-check symbolic work when required.

How to use (3 steps)

  1. Choose 2×2 or 3×3 and pick Equation input or Matrix input. Load a sample from the Examples menu if you want to see a ready-made system.
  2. Enter your equations or the augmented matrix coefficients carefully (one row per equation).
  3. Press Solve to see the solution and, if enabled, elimination steps. Adjust the “Prefer fractional output” and “Show elimination steps” options as needed.

Use this as a learning aid; always write up final work in the format your course requires.

Dimension

FAQ

How does the solver classify the system?

It computes the ranks of A and [A|b] with RREF. If rank(A) = rank([A|b]) = n the solution is unique; if rank(A) = rank([A|b]) < n there are infinitely many solutions; if rank(A) < rank([A|b]) the system is inconsistent.

Can I switch between equation and matrix input?

Yes. Enter equations such as 2x+3y=7 or fill the matrix directly. Examples populate both views and you can share the current setup through the URL.

What should I enter first?

Start with the minimum required inputs shown above the calculate button, then keep optional settings at their defaults for a first pass. After getting a baseline, change one parameter at a time so you can explain which assumption moved the output.

How precise are the results?

The solver keeps exact fractions where possible and rounds only the displayed decimal view. Differences usually come from row-operation order, fraction simplification, or the tolerance used to decide whether a pivot is effectively zero.

Why can my result differ from another calculator?

Compare the augmented matrix, pivot order, and elimination steps first. A different row swap or tolerance can produce a different-looking path while still representing the same solution set.

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