Math & Algebra

Free System of Linear Equations Solver — 2x2 & 3x3 Cramer's Rule

Systems of linear equations are foundational to algebra, physics vector analysis, engineering circuits, and economic equilibrium modeling. Solving multiple equations with multiple unknown variables by hand can lead to tedious algebraic errors. Our free linear equations solver provides instant solutions for 2x2 and 3x3 simultaneous linear equations. Utilize our cramers rule calculator step by step to view complete determinant evaluations (D, Dx, Dy, Dz) and verify exact numerical values for x, y, and z.

System of Linear Equations Solver

Solve 2x2 & 3x3 simultaneous linear equations step-by-step using Cramer's Rule

Enter System Coefficients:

x +y =

Solution Vector Results

Final Solutions:

x = 3, y = 2

Step-by-Step Cramer's Rule Steps:

Main Determinant D = (2 × -1) - (5 × 3) = -17
Determinant Dx = (12 × -1) - (13 × 3) = -51
Determinant Dy = (2 × 13) - (5 × 12) = -34
x = Dx / D = -51 / -17 = 3
y = Dy / D = -34 / -17 = 2

Mathematical Formulas & Calculation Methods

Cramer's Rule for 2x2 Linear System

For system a1 x + b1 y = c1 and a2 x + b2 y = c2 with determinant D = a1b2 - a2b1 ≠ 0:

x = Dx / D = (c1b2 - c2b1) / D | y = Dy / D = (a1c2 - a2c1) / D

3x3 System Cramer's Rule Determinants

Solves 3 equations with 3 unknowns using matrix column substitution.

x = Dx / D, y = Dy / D, z = Dz / D (where D ≠ 0)

Worked Step-by-Step Examples

Example 1: Solving 2x2 System (2x + 3y = 12, 5x - y = 13)

"Find x and y for equations 2x + 3y = 12 and 5x - y = 13."

Steps:

Calculate D = (2 × -1) - (5 × 3) = -2 - 15 = -17.
Calculate Dx = (12 × -1) - (13 × 3) = -12 - 39 = -51.
Calculate Dy = (2 × 13) - (5 × 12) = 26 - 60 = -34.
Compute x = Dx / D = -51 / -17 = 3.
Compute y = Dy / D = -34 / -17 = 2.

Result: x = 3, y = 2

What is System of Linear Equations Solver?

A system of equations calculator evaluates simultaneous linear equations to locate point intersections in 2D or 3D coordinate space.

Cramer's Rule is an explicit theorem in linear algebra that uses determinants to solve systems of linear equations with an equal number of equations and variables.

If the main matrix determinant D equals zero (D = 0), the system has no unique solution (the lines or planes are parallel or infinitely overlapping).

How to Use System of Linear Equations Solver

1Select system dimension: 2x2 (two equations, x & y) or 3x3 (three equations, x, y & z).
2Enter the numeric coefficients into the algebraic equation input grid.
3Review the solution values alongside step-by-step determinant breakdowns.

Key Features

Supports 2x2 and 3x3 simultaneous linear systems.
Step-by-step Cramer's rule determinant breakdowns (D, Dx, Dy, Dz).
Automatic detection of zero-determinant singular systems.
Supports negative and decimal coefficients.

Benefits of Using System of Linear Equations Solver

Ideal for verifying high school algebra and college linear algebra homework.

Saves significant time in physics and electrical engineering mesh analysis.

Clear, readable mathematical notation.

Practical Use Cases

Algebra Students

Check homework steps when using substitution or elimination methods.

Engineering Students

Solve Kirchhoff's circuit node voltage and loop current equations.

Frequently Asked Questions

What happens if determinant D equals zero?

If D = 0, Cramer's rule cannot be applied because division by zero is undefined. The system either has no solutions (parallel lines) or infinitely many solutions (identical lines).

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