Home › Calculators › System of Equations Solver

System of Equations Solver

Type your linear equations as you would write them, such as 2x + 3y − z = 5, one per line. The solver finds exact answers as fractions and shows Gaussian elimination or Cramer's rule step by step.

Equations (one per line)

Use any single letters as unknowns. Coefficients can be decimals or fractions: 0.5x, 2/3y. Constants may be on either side.

Show working using

Gaussian eliminationCramer's rule

Enter your values to see the result.

How to use the System of Equations Solver

Type one equation per line, e.g. 2x + 3y − z = 5.
Use decimals or fractions as coefficients if you need them (0.5x, 2/3y).
Choose the method for the working: Gaussian elimination or Cramer's rule.
Press Solve: the answer also updates as you type.
Read the solution and follow the steps and the check.

Methods for solving linear systems

Write the system as A·x = b, where A holds the coefficients, x the unknowns and b the constants.

Cramer's rule: x_i = det(A_i) / det(A), where A_i is A with column i replaced by b. For a 2×2 system ax + by = e, cx + dy = f: x = (ed − bf) / (ad − bc) and y = (af − ec) / (ad − bc).

Gaussian elimination: use row operations on the augmented matrix [A | b] to create zeros below and above each pivot, divide each pivot row so the pivot is 1, and read the solution from the last column.

Example: 3x + 2y = 16 and x − y = 2. D = 3·(−1) − 2·1 = −5, D_x = 16·(−1) − 2·2 = −20, D_y = 3·2 − 16·1 = −10, so x = −20 / −5 = 4 and y = −10 / −5 = 2.

Types of solutions

Case	Determinant	Geometry (2 unknowns)	Example
One unique solution	≠ 0	Lines cross at one point	x + y = 3, x − y = 1 → (2, 1)
No solution	= 0	Parallel lines	x + 2y = 4, 2x + 4y = 9
Infinitely many	= 0	The same line twice	x + 2y = 4, 2x + 4y = 8

With three unknowns each equation is a plane: a unique solution is a single point where all three planes meet.

Frequently asked questions

How do you solve a system of linear equations?

Common methods are substitution, elimination, Gaussian elimination on the augmented matrix and Cramer's rule. All give the same answer when the system has a unique solution.

What is Cramer's rule?

For a square system Ax = b with det(A) ≠ 0, each unknown equals Dᵢ / D, where D is the determinant of A and Dᵢ is the determinant of A with column i replaced by b.

What is Gaussian elimination?

It applies row operations (swap rows, scale a row, add a multiple of one row to another) to the augmented matrix until it is in reduced row echelon form, from which the solution can be read directly.

What does it mean if the determinant is zero?

The system has no unique solution: either it has no solution at all (the equations contradict each other) or infinitely many (one equation is a combination of the others). The solver tells you which.

Can I use fractions and decimals?

Yes. Coefficients like 0.25x or 3/4y are converted to exact fractions and all arithmetic is exact, so answers such as x = 7/3 are not rounded.

How many equations do I need?

For a unique solution you need as many independent equations as unknowns: 2 for x and y, 3 for x, y and z.

Related tools