Elimination Calculator 3x3

Enter 3 Equations

a11 coefficient

a12 coefficient

a13 coefficient

Equation 1 right side

a21 coefficient

a22 coefficient

a23 coefficient

Equation 2 right side

a31 coefficient

a32 coefficient

a33 coefficient

Equation 3 right side

Decimal precision

Zero tolerance

Result format

Partial pivoting

Example Data Table

Use this sample to test the calculator. The solution is x = 2, y = 3, and z = -1.

Equation	x coefficient	y coefficient	z coefficient	Right side
1	2	1	-1	8
2	-3	-1	2	-11
3	-2	1	2	-3

Formula Used

The calculator solves this system:

a11x + a12y + a13z = d1 a21x + a22y + a23z = d2 a31x + a32y + a33z = d3

It builds the augmented matrix [A | b]. Then it applies elimination.

Ri = Ri - (aik / akk)Rk

After upper triangular form is reached, it uses back substitution:

z = u34 / u33 y = (u24 - u23z) / u22 x = (u14 - u12y - u13z) / u11

The determinant check is:

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

If det(A) ≠ 0, the system has one unique solution.

How to Use This Calculator

Enter the coefficients for x, y, and z.
Enter the right side value for each equation.
Select decimal precision and result format.
Keep partial pivoting enabled for better stability.
Click the calculate button.
Review the solution, determinant, ranks, graph, and steps.
Download the CSV or PDF file if needed.

About the 3x3 Elimination Method

A 3x3 elimination calculator helps solve three linear equations with three unknowns. It changes the system into a simpler upper triangular form. Then it uses back substitution to find x, y, and z. The method is fast. It is also easy to audit because every row operation is visible.

Why Elimination Works

Each equation represents a plane in three dimensional space. The solution is the point where the planes meet. Elimination keeps the same solution set while replacing equations with easier versions. A row swap changes equation order. A row addition removes one coefficient. A row scale changes equation size, not its meaning.

Advanced Result Review

This calculator checks more than the final numbers. It reports the determinant, ranks, residual errors, and pivot actions. A nonzero determinant means one unique solution exists. A zero determinant needs more checking. The system may have no solution. It may also have infinitely many solutions.

Practical Uses

3x3 systems appear in budgeting, mixtures, forces, chemistry, electrical networks, and business allocation problems. They are also common in classroom algebra and engineering worksheets. Manual work can be slow. A small sign mistake can change the result. This page reduces that risk by showing the operations and verification.

Better Accuracy

Partial pivoting is included because it improves numerical stability. It chooses a stronger pivot before eliminating entries below it. This is useful when a pivot is small. The tolerance setting controls how close a value can be to zero before the calculator treats it as zero. More precision gives longer decimal answers. Less precision gives cleaner summaries.

Reading the Graph

The Plotly chart draws equation planes when possible. The solution point is marked when the system has one answer. The graph is a visual aid. The numeric steps remain the main source for exact checking.

Exporting Your Work

Use CSV export for spreadsheets. Use PDF export for printing or sharing. Both downloads include the equations, options, result summary, solution, determinant, and residual checks. Keep them with class notes, project records, or client calculations. For best results, enter equations in the same unit system. Check signs before solving. Review residuals after every calculation.

FAQs

1. What does this 3x3 elimination calculator solve?

It solves three linear equations with three unknowns. It finds x, y, and z when a unique solution exists. It also detects inconsistent and dependent systems.

2. What is Gaussian elimination?

Gaussian elimination uses row operations to convert a system into upper triangular form. Then back substitution gives the unknown values.

3. Why is partial pivoting useful?

Partial pivoting chooses the largest available pivot in a column. This can reduce rounding error and avoid weak pivots.

4. What does a zero determinant mean?

A zero determinant means there is no unique solution. The system may have no solution or infinitely many solutions.

5. What are residuals?

Residuals compare the calculated left side with the right side. Values near zero show the answer fits the original equations well.

6. Can I use decimal coefficients?

Yes. You can enter integers, decimals, negative numbers, and fractional decimal values. Use enough precision for sensitive systems.

7. What does the graph show?

The graph shows the equation planes when they can be drawn as z functions. The solution point appears for a unique solution.

8. What should I export?

Use CSV for spreadsheet work. Use PDF when you need a printable report with equations, results, and checks.