EN

System of Equations Calculator

Type the equations just as they appear in your book. The system of equations calculator solves them exactly by Gaussian elimination and shows every row operation.

Source: Wolfram MathWorld – Gaussian elimination. Updated: .

Variables as letters (x, y, z … or x1, x2); numbers like 0.5 or 3/4; terms may appear on both sides.
My ToolboxYour inputs are saved in this browser only.

Result

Solution
x = 3; y = 2
Solution set
exactly one solution
Determinant of the coefficient matrix
−5
Steps (Gauss-Jordan)
  • Augmented matrix: (2 3 | 12) (1 −1 | 1)
  • R1 : 2 → (1 3/2 | 6) (1 −1 | 1)
  • R2 − R1 → (1 3/2 | 6) (0 −5/2 | −5)
  • R2 : (−5/2) → (1 3/2 | 6) (0 1 | 2)
  • R1 − (3/2) · R2 → (1 0 | 3) (0 1 | 2)
Check
  • 2x + 3y = 12: 12 = 12 ✓
  • x − y = 1: 1 = 1 ✓
Rank = number of variables = 2, so there is exactly one solution.

How it is calculated

A system of linear equations has exactly one solution, no solution, or infinitely many solutions. Gaussian elimination tells you which: it turns the system into row echelon form step by step without changing the solution set.

Gaussian elimination

  1. Write the coefficients and right-hand sides as an augmented matrix – 2x + 3y = 12 and x − y = 1 become (2 3 | 12) and (1 −1 | 1).
  2. Swap rows, divide a row by a number, or subtract a multiple of one row from another, until there are zeros below and above the diagonal.
  3. When the left side is the identity matrix, the right side is the solution: x = 3, y = 2.

The calculator works with exact fractions, so there are no rounding errors, and it substitutes the answer back into every original equation as a check.

Other methods

For two equations, school courses also use substitution (solve one equation for x and substitute), elimination (add or subtract the equations to remove a variable) and graphing (the solution is where the lines cross). All give the same answer; elimination is exactly the Gaussian step “subtract a multiple of a row”. For square systems, Cramer’s rule uses determinants – if the determinant is 0, there is no unique solution.

No solution or infinitely many?

If a row becomes 0 = 5, the equations contradict each other: no solution (two parallel lines). If a row becomes 0 = 0, one equation was redundant; some variables stay free and there are infinitely many solutions. Single quadratic equations are solved by the quadratic formula calculator, fractions by the fraction calculator, and any expression by the scientific calculator.

Frequently asked questions

How do you solve a system of two equations?

For example by elimination: 2x + 3y = 12 and x − y = 1. Multiply the second by 3 and add: 5x = 15, so x = 3 and y = 2.

What is Gaussian elimination?

A method that uses row operations to bring a linear system into row echelon form, after which the variables can be read off one by one.

When does a system have no solution?

When a row 0 = c with c ≠ 0 appears. Then the augmented matrix has a higher rank than the coefficient matrix.

When are there infinitely many solutions?

When the system is consistent but has fewer independent equations than variables. The remaining variables are free parameters.

Can I enter fractions and decimals?

Yes, e.g. 0.5x + 3/4y = 2. The calculator works exactly with fractions and shows the answer as a fraction and a decimal.

Sources and legal basis

As of: · Editorial policy

Related tools