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System of Equations Solver

Solve systems of linear equations (2x2, 3x3) with step-by-step logic. Ideal for algebra students and engineering models.

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System Equation Solver

Solve simultaneous linear equations in 2 or 3 variables instantly using matrix determinant methods.

Input Matrix [A | B]

x
y
=
x
y
=

Current Equation Representation:

1x + 1y = 5
1x + -1y = 1

Solution Output

x value3.00
y value2.00

Mathematical Proof (Cramer's Rule)

1

Determinant (D) = (1 * -1) - (1 * 1) = -2.00

2

Dx = (5 * -1) - (1 * 1) = -6.00

3

Dy = (1 * 1) - (5 * 1) = -4.00

4

x = Dx / D = 3.00

5

y = Dy / D = 2.00

How it works

This solver uses **Cramer's Rule**, a method for solving systems of linear equations using determinants. It calculates the main determinant of the coefficient matrix and then substitute each column with the constant vector to find variable-specific determinants.

Constraints

A system has a unique solution if and only if the determinant is **non-zero**. If the determinant is zero, the system is either inconsistent (no solution) or dependent (infinitely many solutions).

This maths solver is part of our broader maths category designed for Algebra Students, Civil Engineers, and Economists (and for everyone else who needs to solve multiple simultaneous linear equations with precision).

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Mastering Simultaneous Equations: The System Solver

A system of equations is a collection of two or more equations with a same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system. Our System of Equation Solver is a high precision maths solver designed to handle 2x2 and 3x3 linear systems using rigorous algebraic methods.

Whether you are calculating the intersection of two lines in geometry or modeling the load distribution in a bridge truss, simultaneous equations are the critical tool. As defined by Britannica, these systems form the basis of linear algebra.

Methods of Solving Linear Systems

Our solver utilizes several classic mathematical approaches to ensure accuracy and provide educational clarity:

  • The Substitution Method: Solving one equation for one variable and substituting it into the other. Ideal for simple 2x2 systems.
  • The Elimination Method: Adding or subtracting equations to eliminate one variable. Efficient for larger systems.
  • Cramer's Rule (Matrix Method): Using determinants to solve for variables. This method is the "gold standard" for 3x3 systems and is frequently used by researchers at NIST for high precision modeling.

Consistent vs. Inconsistent Systems

When using this maths category tool, you may encounter different types of results:

Unique Solution

The lines/planes intersect at a single point. The system is consistent and independent.

No Solution

The lines are parallel and never meet. The system is inconsistent.

Infinite Solutions

The equations describe the same line/plane. The system is consistent and dependent.

Real World Applications

In the professional world, systems of equations are used to solve complex optimization problems:

  • Electrical Engineering: Applying Kirchhoff's laws to find the current and voltage in different branches of a circuit.
  • Economics: Finding the "Market Equilibrium" where the supply equation and demand equation meet.
  • Chemistry: Balancing chemical equations to ensure the conservation of mass.
  • Navigation: GPS systems solve systems of non-linear equations (approximated as linear) to determine your exact latitude and longitude.

Educational Excellence

Mastering these systems is a prerequisite for advanced mathematics and physics. Leading universities like MIT dedicate entire semester-long courses to the study of linear systems (Linear Algebra). Our tool provides a practical environment to test your homework and build your "computational intuition."

Inputting Your Matrix

To use the System Solver, enter the coefficients (a, b, c) and the constant (d) for each equation in the format: ax + by + cz = d. Our interface will automatically organize these into a matrix and compute the result.

Pro Tip: Always double-check your signs (+/-). A single sign error is the most common reason for getting an incorrect solution in simultaneous equations.

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