Linear System Generator

What this generator does

Draws a coefficient matrix, rejects it if the determinant is zero, then chooses a whole-number solution and computes the constants from it. Showing the answer also shows the elimination: every row operation named, and the grid it produced printed under it, down to the identity where the answer can simply be read off.

How to use this tool

1. Choose how many unknowns.
2. Solve by elimination, or by Cramer's rule if you prefer determinants.
3. Check your answer by substituting back into every equation.
4. Reveal the solution to confirm.

Understanding the controls

Matrix size

Between 2 and 5 unknowns. Five is where elimination by hand starts to be a genuine exercise in bookkeeping.

Seed

Reproduces exactly the same system, so it can be set and marked later.

Show the answer

Reveals the value of every unknown and the elimination that reaches it, so the system can be set as an exercise, marked afterwards, and — more useful — compared line by line against what a student actually wrote.

Common use cases

• Elimination practice beyond the two-unknown case
• Cramer's rule exercises with a stated determinant
• Test systems for a solver, with the answer known
• Showing why a non-zero determinant is what guarantees one solution
• Setting the same system for a group from a seed

How this generator works

Because the solution is chosen first and the constants derived from it, the answer is exact and always whole. The check substitutes the solution back into every equation and requires each to balance exactly, and confirms the determinant is non-zero so the solution really is the only one. The working is checked separately and more strictly: every line is re-done by applying the operation it names to the grid above it and comparing, and the original answer is confirmed to satisfy every row of every intermediate grid — because working that reaches the right answer through a wrong line has still taught somebody something false. Every number in it is an exact fraction rather than a decimal.

Randomness and fairness

The coefficients and the solution are random; the constants follow from them. Seeded systems reproduce and are therefore explicitly not cryptographically secure, and unseeded ones use the browser's cryptographically secure generator.

For how randomness is produced across the whole site, see how Generate Random works.

Limitations and good to know

• Square systems only — over- and under-determined systems have different answers entirely.
• Solutions are always whole numbers, which is unrealistic but is what makes the arithmetic checkable.
• The working is Gauss—Jordan elimination, which ends at the identity so the answer can be read off. Cramer's rule, LU decomposition and back-substitution from a triangular form are all valid and none of them is what is shown.
• Coefficients can be zero, which occasionally makes a system easier than its size suggests.
• A system disappears when you leave; seed it to set the same one twice.

Privacy and your data

Coefficients and solutions are computed in your browser. Nothing about the system or your seed leaves the page.