Simultaneous Equations Solver (Matrix Method, 2 to 8 Unknowns)

Solves n equations in n unknowns by Gaussian elimination and shows every row operation, with a check at the end.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Write each equation as coefficients followed by the number on the right of the equals sign, one equation per line. The calculator builds the augmented matrix, picks the largest number in each column as the pivot (partial pivoting keeps rounding error small), clears the entries below it, then back-substitutes from the last row up.

If the equations are dependent or inconsistent there is no single answer, and the calculator says so. Typical uses are statics (reactions of several unknowns), circuits (mesh and node equations), and fitting curves.

Formulas used

  • [A\,|\,b]\ \to\ \text{upper triangular}\ \to\ \mathbf{x}
  • \max|A\mathbf{x} - \mathbf{b}|\ \text{(check)}

Worked example

With the default numbers in the calculator above:

  1. The equations as an augmented matrix: 4 equations, 4 unknowns
  2. Column 1: largest pivot to the top, then clear below it: pivot 3
  3. Column 2: largest pivot to the top, then clear below it: pivot 3.333
  4. Column 3: largest pivot to the top, then clear below it: pivot -2.5
  5. Column 4: largest pivot to the top, then clear below it: pivot 3.2
  6. Back substitution, from the last row upwards: x_4 first
  7. Solution: x1 = 1, x2 = -1, x3 = 2, x4 = 3
  8. Check: put the answers back into the original equations: all equations satisfied

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

How do I enter an equation?

Rewrite it as a₁x₁ + a₂x₂ + ... = b. The line is then a₁ a₂ ... b. Use 0 for a missing unknown.

What is partial pivoting?

Swapping rows so the largest number in the column is the pivot. It avoids dividing by a tiny number.

Why does it say there is no single answer?

The equations are dependent (one is a combination of the others) or contradict each other.

Related calculators

Further reading: Any linear algebra text, for example Strang, Introduction to Linear Algebra.

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