How to use the Gauss Elimination Method Calculator
- 1. Enter the equationsOne equation per line: coefficients then the right-hand side, like 2 1 1 10.
- 2. Choose optionsTurn on partial pivoting if your syllabus asks for it, and pick fractions or decimals.
- 3. Read each stepEvery row operation is listed with the matrix after it, then the back substitution.
How Gauss elimination works
Write the system as the augmented matrix [A | b]. Use the first equation to make every entry below the first pivot zero with Rᵢ → Rᵢ − (aᵢ₁/a₁₁)R₁. Repeat with the second pivot, and so on, until the matrix is upper triangular. The last equation then has one unknown; solve it and substitute upwards (back substitution).
If a pivot is zero, swap in a lower row. With partial pivoting the row with the largest pivot is always chosen, which reduces rounding error in decimal work. The tool keeps exact fractions, so its answer has no rounding error at all.
Worked example
Solve 2x + y + z = 10, 3x + 2y + 3z = 18, x + 4y + 9z = 16. Step 1: R₂ → R₂ − (3/2)R₁ and R₃ → R₃ − (1/2)R₁ give rows [0, 1/2, 3/2 | 3] and [0, 7/2, 17/2 | 11]. Step 2: R₃ → R₃ − 7R₂ gives [0, 0, −2 | −10].
Back substitution: z = −10/−2 = 5; y = (3 − (3/2)·5)/(1/2) = −9; x = (10 + 9 − 5)/2 = 7. So x = 7, y = −9, z = 5. The product of the pivots, 2 × 1/2 × (−2) = −2, is the determinant of A.
No solution or infinitely many
If a row becomes [0 0 0 | c] with c ≠ 0, the equations are inconsistent and there is no solution. If a whole row becomes zero, rank(A) is less than the number of unknowns and there are infinitely many solutions. The tool reports both cases. To see the reduced form directly, use the Gauss-Jordan calculator.
Frequently asked questions
What is the difference between Gauss elimination and Gauss-Jordan?
Gauss elimination stops at an upper triangular matrix and uses back substitution. Gauss-Jordan also clears the entries above each pivot, so the answer can be read directly.
Why use partial pivoting?
Dividing by a small pivot magnifies rounding errors in decimal arithmetic. Choosing the largest available pivot keeps the multipliers at most 1 in size.
Can I enter fractions?
Yes. Use entries like 3/4, −1/2 or 0.25. The working stays in exact fractions unless you choose decimals.
How many equations can it solve?
From 2 to 6 equations with the same number of unknowns.
Is this calculator free?
Yes. You fill a short form once when you see your first result, and other Uncodemy tools then open without the form. Everything is calculated in your browser.