System of Equations Calculator

Solve systems of 2 or 3 linear equations using Cramer's rule, showing determinants, solution type, and step-by-step work.

Last updated 7 March 2026

System size

Equation 1

x + y =

Equation 2

x + y =

Solution

x = 2.2, y = 1.2

The system has a unique solution: x = 2.2, y = 1.2.

Solution Type: Unique
Determinant (D): -5
x: 2.2
y: 1.2

Step-by-Step Solution

System: 2x + 3y = 8

1x + -1y = 1

Coefficient determinant D = (2)(-1) - (3)(1) = -5

Dx = (8)(-1) - (3)(1) = -11

Dy = (2)(1) - (8)(1) = -6

x = Dx / D = -11 / -5 = 2.2

y = Dy / D = -6 / -5 = 1.2

Interpretation

The determinant D = -5 is non-zero, confirming a unique solution exists. The equations intersect at exactly one point.

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Algebra

System of equations calculator: solve linear systems with Cramer's rule

A system of equations calculator solves two or three simultaneous linear equations for their unknown variables. It uses Cramer's rule (determinant-based method) to find exact solutions and identifies whether the system has a unique solution, infinitely many solutions, or no solution at all.

Cramer's rule for 2×2 systems

For a 2×2 system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, the determinant D = a₁b₂ − a₂b₁. When D ≠ 0, the system has a unique solution: x = (c₁b₂ − c₂b₁)/D and y = (a₁c₂ − a₂c₁)/D.

If D = 0, the system is either inconsistent (no solution — parallel lines) or dependent (infinitely many solutions — the same line).

D = a₁b₂ − a₂b₁

Determinant of the coefficient matrix. This is the specific relationship the calculator applies when building the result.

x = (c₁b₂ − c₂b₁) / D

Solution for x using Cramer's rule. This is the specific relationship the calculator applies when building the result.

Worked example and interpretation

A worked example helps translate the system of equations calculator maths into a realistic scenario so the user can compare the headline result with a concrete set of inputs.

That matters because a result is easier to trust when the page shows how the same logic behaves in a practical case instead of leaving the formula abstract.

Using the result well

Use the system of equations calculator output as a planning aid, then compare it with the assumptions, units, and caveats shown elsewhere on the page before acting on the number alone.

That extra interpretation step matters because a calculator can simplify the arithmetic but still cannot replace real-world context such as local rules, contract terms, or individual circumstances.

Frequently asked questions

What does it mean when the determinant is zero?

A zero determinant means the equations are either parallel (no solution) or identical (infinitely many solutions). You cannot divide by zero in Cramer's rule, so the method signals that a unique solution does not exist.

Can this solve non-linear systems?

No — Cramer's rule and the methods used here apply only to linear equations (variables to the first power with no products of variables). Non-linear systems require different techniques like substitution or numerical methods.

How can I check the system of equations calculator: solve linear systems with cramer's rule result manually?

The safest manual check is to follow the same formula or rule one step at a time and compare that working with the calculator output. That catches sign errors, bracket mistakes, and input-order mixups without requiring any extra method beyond the underlying maths itself.