How to use the gaussian elimination calculator
Prepare the values for this calculation
Solve a 3×3 linear system by Gaussian elimination with partial pivoting. Use the formula below to check that it answers your question. Read the input units before transferring figures from a statement, measurement or dataset.
Enter and check each input
- a₁₁: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₁₂: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₁₃: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- b₁: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₂₁: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₂₂: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₂₃: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- b₂: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₃₁: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₃₂: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- a₃₃: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
- b₃: enter a finite number. Use the unit named in the label or selected above. Do not leave this required field blank.
The input table gives a complete example. Keep units and time periods consistent across related fields.
Calculate and resolve any input error
Choose Calculate after entering the values. Changes also update the answer after a short pause. If an error appears, follow its message, correct the field and calculate again. Reset restores the example values.
Interpret the answer
Check Solution first. With the unchanged example inputs, it is x = 1.2857142857, y = 2.1428571429, z = 2.5714285714. Interpret it using the formula and assumptions on this page.
Compare and keep a useful result
For a controlled comparison, keep every other example input unchanged and change a₁₁ from 1 to 1.1. The resulting Solution is x = 1.3043478261, y = 2.0869565217, z = 2.4782608696. This illustrates the equation’s response to that input, rather than a forecast of real events.
The result diagram shows the inputs, method and output. A numeric chart is shown only when the calculation supports a meaningful numeric relationship.
Use Copy result or Download CSV to keep the answer. Share tool shares the page address without placing your input values in the URL. Recheck inputs and assumptions before using the result for a decision.
Gaussian Elimination Calculator formula
Row-reduces the augmented matrix to solve a unique linear system.
Inputs and units
| Input | Format / range | Example |
|---|---|---|
| a₁₁ | Number | 1 |
| a₁₂ | Number | 1 |
| a₁₃ | Number | 1 |
| b₁ | Number | 6 |
| a₂₁ | Number | 2 |
| a₂₂ | Number | -1 |
| a₂₃ | Number | 1 |
| b₂ | Number | 3 |
| a₃₁ | Number | 1 |
| a₃₂ | Number | 2 |
| a₃₃ | Number | -1 |
| b₃ | Number | 3 |
Worked example
Using the example inputs above, the solution is x = 1.2857142857, y = 2.1428571429, z = 2.5714285714.
- Solution
- x = 1.2857142857, y = 2.1428571429, z = 2.5714285714
Results and assumptions
The output uses the solution convention and formula shown above. Match the entered units and source assumptions when comparing results.
Fix an input or unexpected result
a₁₁ is required. Enter a₁₁ using the format and bounds in the input table. Zero and an empty field are different: use zero only if it is allowed and represents your data.
If your answer differs from the example, choose Reset and compare every field, including selected units and options. Check percentage versus decimal entry and whether values describe the same period. Round only the final displayed answer when reproducing the formula.
If a discrepancy remains, report the calculator issue with the page address, non-sensitive example inputs, actual result, expected result and a reference for your calculation.