Simultaneous Equations Generator
A pair of equations in x and y, generated so that the answer is always a whole number pair. The construction runs the other way from the solving: a solution is picked first, then coefficients, and the constants are whatever makes those coefficients true of that solution. The pair is rejected if its determinant is zero, because two parallel lines or the same line written twice do not have the single answer the question promises.
What this generator does
Produces pairs of equations of the form ax + by = e and cx + dy = f, where x and y are whole numbers you choose the range for. The working shown multiplies each equation to match the y coefficients, subtracts to eliminate, and substitutes back — the standard elimination route, with the actual numbers filled in.
How to use this tool
- Choose how many pairs you want and how large the numbers should get.
- Work each one through by elimination or substitution.
- Reveal the answers, or the full working, to check.
- Seed the sheet if a whole class should get the identical set.
Understanding the controls
- How many questions
- Between 1 and 20. Ten is about a page once the working is shown.
- Level
- Gentle keeps solutions and coefficients within four, standard within eight, stretch within twelve — so the method stays the same and only the arithmetic load changes.
- Seed
- Any word reproduces the identical sheet, which is how you give two people the same questions without saving a file.
- Show the answers / working
- Answers alone for marking, or the full elimination for teaching.
Common use cases
- A homework sheet on elimination that never runs out of questions
- Starter questions where the arithmetic stays small enough to do mentally
- Practice at the point where substitution beats elimination and the other way round
- Setting two pupils the same questions from a shared seed
- Checking a solver or a spreadsheet against systems whose answer is known
How this generator works
The solution is drawn first, then four coefficients. Any draw whose determinant ad − bc is zero is rejected and redrawn, since that pair is either inconsistent or has infinitely many solutions. The constants follow from substituting the solution. Before display, the solution is substituted back into both equations and the determinant re-checked.
Randomness and fairness
Which solution and coefficients you get is random; that both equations balance is not. Seeded sheets are reproducible and therefore explicitly not cryptographically secure. Without a seed the browser's cryptographically secure generator is used.
For how randomness is produced across the whole site, see how Generate Random works.
Limitations and good to know
- Both equations are linear in two unknowns. Three-unknown systems and non-linear pairs are not offered.
- Solutions are always whole numbers, which is deliberate for practice but is not what a real system usually gives you.
- The working shown is elimination. Substitution and matrix methods reach the same answer by a different route that is not written out.
- Nothing is marked — the answers are revealed for you to compare against, not entered.
- Sheets are not stored, so seed anything you intend to print or reissue.
Privacy and your data
Questions are constructed in your browser. Your seed, level and the questions themselves are never transmitted and never kept between visits.
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