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Last updated: July 31, 2026

Gauss-Jordan Elimination Calculator

Dharmendra Singh - Creator

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Balram Rajput
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Formula

For a1x+b1y=c1 and a2x+b2y=c2: det=a1b2-a2b1, x=(c1b2-c2b1)/det, y=(a1c2-a2c1)/det

Where:

  • a1=a1 coefficient
  • b1=b1 coefficient
  • c1=c1 constant
  • a2=a2 coefficient
  • b2=b2 coefficient
  • c2=c2 constant
Gauss-Jordan EliminationAn augmented matrix being row-reduced to identity form, revealing the solution values x and y directly.[ 1 1 | 3 ][ 1 -1 | 1 ]row-reduce[ 1 0 | 2 ][ 0 1 | 1 ]x = 2, y = 1
Gauss-Jordan elimination reduces the augmented matrix to identity form, reading the solution directly from the last column.

Worked Examples

Unique system

Solve 2x+3y=7 and x-y=1.

  1. 1det = 2(-1)-1(3) = -5
  2. 2x = (7(-1)-1(3))/(-5)=2
  3. 3y = (2(1)-1(7))/(-5)=1
Final Answer: x=2, y=1, determinant=-5

Fractional solution

Solve x+2y=5 and 3x-y=4.

  1. 1det = 1(-1)-3(2)=-7
  2. 2x = (5(-1)-4(2))/(-7)=13/7
  3. 3y = (1(4)-3(5))/(-7)=11/7
Final Answer: x=1.857143, y=1.571429, determinant=-7

Negative coefficients

Solve -2x+y=3 and 4x+5y=6.

  1. 1Compute determinant -14
  2. 2Apply formulas for x and y
  3. 3Round to 6 decimals
Final Answer: x=-0.642857, y=1.714286, determinant=-14

Introduction

Gauss-Jordan elimination for 2×2 systems finds the unique intersection point when the determinant is non-zero.

Formula and Method

Gauss-Jordan Elimination Calculator applies For a1x+b1y=c1 and a2x+b2y=c2: det=a1b2-a2b1, x=(c1b2-c2b1)/det, y=(a1c2-a2c1)/det and returns deterministic outputs from validated inputs.

  • Validate each required input

  • Apply the stated formula

  • Round numeric outputs where needed

  • Return safe defaults when validation fails

Input Fields

Use these keys exactly as defined in calculation.ts.

  1. 1

    a1: a1 coefficient

  2. 2

    b1: b1 coefficient

  3. 3

    c1: c1 constant

  4. 4

    a2: a2 coefficient

  5. 5

    b2: b2 coefficient

  6. 6

    c2: c2 constant

Output Fields

These values map directly to calculation.ts return keys.

  1. 1

    x: Solved x value

  2. 2

    y: Solved y value

  3. 3

    determinant: det = a1b2-a2b1 for uniqueness check

Validation Rules

All coefficients must be finite and determinant must be non-zero for a unique solution.

  • Finite numeric values are required

  • Domain restrictions are enforced before solving

  • Invalid input returns default-safe outputs

  • Check examples to confirm expected behavior

How to Use This Calculator

Enter inputs, run the calculation, and read the primary output first.

  • Enter all required values

  • Click calculate

  • Review primary output first

  • Use supporting outputs for interpretation

Practical Uses

This calculator is useful for study, verification, and fast applied math checks.

  • Linear algebra practice

  • Circuit and force-balance equations

  • Intersection problems

  • Quick equation verification

FAQs

What does the Gauss-Jordan Elimination Calculator compute?

It computes x Solution and supporting outputs from your inputs.

Which inputs are required?

Required inputs are: a1 coefficient, b1 coefficient, c1 constant, a2 coefficient, b2 coefficient, c2 constant.

How are invalid inputs handled?

When validation fails, the calculator returns safe default values and does not attempt invalid math.

Are results rounded?

Yes. Numeric outputs are rounded inside calculation.ts (typically to 6 decimal places).

Can I use negative or decimal values?

Finite real coefficients are accepted when they define a unique solution.

How should I verify my answer?

Use the worked examples and compare each output key with your manual steps.

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