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Math Solvers/Algebra & Calculus/linear-equations-system

System of Linear Equations Solver (2×2 & 3×3)

Interactive simultaneous equation solver supporting both 2-variable and 3-variable linear systems with full determinant expansion and consistency checks.

Linear Equations (a₁x + b₁y = c₁)

Equation 1
Equation 2
Value of x
1.0000
Value of y
2.0000
Determinants:
D = -5.0000, Dx = -5.0000, Dy = -10.0000

Cramer's Rule Solution Steps

1. System Representation in Matrix Form

Linear system with coefficient matrix A and constant vector B.

2. Main Coefficient Determinant (D)

D ≠ 0 confirms a single unique solution exists.

3. Component Determinants (Dx and Dy)

Computed by replacing x and y coefficient columns with constants vector.

4. Cramer's Rule Solution

Unique intersection point (x, y) = (1.0000, 2.0000).

Recommended Handbooks & Professional Tools

Essential high-yield reference standards and hardware for System of Linear Equations Solver (2×2 & 3×3)

Mechanical & ManufacturingIndustry Bible ($120+)

Machinery's Handbook (31st Edition - Toolbox Edition)

by Erik Oberg, Franklin D. Jones

The 3,000-page ultimate manufacturing & mechanical engineering reference.

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Civil & StructuralEssential Code ($300+)

AISC Steel Construction Manual (16th Edition)

by American Institute of Steel Construction

The official code manual for structural steel design, member tables & specifications.

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Curated for practicing engineers, FE/PE candidates, and university researchers.* As an Amazon Associate, we earn from qualifying purchases at no additional cost to you.

Mathematical Formulation

Standard analytical and numerical formulations derived from classical algebra, linear algebra, and calculus principles.

Engineering Reference & Technical Guide

Systems of Linear Equations & Cramer's Rule Reference Guide

Peer-reviewed mathematical formulation, verified step-by-step worked engineering calculations, design codes, and practical guidelines.

Governing Theory & Derivation

Analytical physics and classical mechanics foundations

Linear systems represent fundamental mathematical modeling in structural truss analysis (method of joints), electrical circuit loop/mesh equations (Kirchhoff's Laws), and fluid network pipe networks. A system of n equations in n unknowns is expressed in matrix notation as A·x = b.

1. Matrix Representation and Determinant

For a 2×2 or 3×3 square coefficient matrix A, the existence of a unique solution requires det(A) ≠ 0:

2. Cramer's Rule Formulation

Each unknown x_i is computed as the ratio of determinants, where A_i is obtained by replacing column i with the constant vector b:

Engineering Interpretation:If det(A) = 0, the system is singular. It either possesses infinitely many solutions (dependent redundant equations) or no solution (inconsistent contradictory constraints).

Step-by-Step Worked Example

Two-Loop DC Electrical Circuit Mesh Current Analysis

Problem Statement

Mesh analysis of a two-loop DC circuit produces the simultaneous linear equations: 12·I_1 - 4·I_2 = 24 and -4·I_1 + 10·I_2 = -8. Solve for mesh currents I_1 and I_2 using Cramer's rule.

Given Parameters

Equation 112·I_1 - 4·I_2 = 24 Volts
Equation 2-4·I_1 + 10·I_2 = -8 Volts

Detailed Calculation Steps

1Compute System Determinant (D)

Since D ≠ 0, a unique set of mesh currents exists.

2Compute D_1 and Solve for I_1

Replacing column 1 with the source voltage vector.

3Compute D_2 and Solve for I_2

Mesh current 2 is exactly zero.

Conclusion & Design Verification:Mesh current I_1 = 2.0 Amperes and I_2 = 0.0 Amperes. The shared 4-ohm resistor carries 2.0 A.

Applicable Standards & Design Codes

Recognized engineering guidelines and criteria

IEEE Std 519IEEE

Recommended Practice and Requirements for Harmonic Control in Electric Power Systems

Employs linear systems for nodal admittance matrix solutions.

Assumptions & Practical Limitations

Critical constraints for professional applications

  • Cramer's rule is computationally inefficient for large systems (n > 4). Gaussian elimination or LU decomposition should be used for large matrices.

Frequently Asked Questions

Common technical engineering inquiries

Q:When does Cramer's rule fail?

Cramer's rule fails when the determinant of the coefficient matrix is zero (det A = 0), indicating parallel planes or linearly dependent equations.