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Simultaneous Equations Calculator

Solve systems of linear equations with detailed step-by-step solutions using multiple mathematical methods

Simultaneous Equations Calculator

Enter Coefficients

x +y =
x +y =

Understanding Simultaneous Equations

What are Simultaneous Equations?

Simultaneous equations are a set of equations with multiple unknowns that are solved together. The solution must satisfy all equations in the system simultaneously.

Example 2×2 System:

2x + 3y = 7

x - y = 1

Types of Solutions

Unique Solution

One specific set of values satisfies all equations

Infinite Solutions

Equations are dependent (same line/plane)

No Solution

Equations are inconsistent (parallel lines)

Solution Methods

Substitution Method

  1. 1. Solve one equation for one variable
  2. 2. Substitute into the other equation
  3. 3. Solve for the remaining variable
  4. 4. Back-substitute to find the first variable

Best for: Simple coefficients

Elimination Method

  1. 1. Multiply equations to align coefficients
  2. 2. Add or subtract equations
  3. 3. Eliminate one variable
  4. 4. Solve for the remaining variable

Best for: Integer coefficients

Matrix Method

  1. 1. Write in matrix form Ax = b
  2. 2. Find the inverse of matrix A
  3. 3. Multiply: x = A⁻¹b
  4. 4. Calculate the solution vector

Best for: Large systems

Worked Examples

2×2 System Example

Given System:

3x + 2y = 12

x - y = 1

Step 1: From equation 2: x = y + 1

Step 2: Substitute into equation 1:

3(y + 1) + 2y = 12

3y + 3 + 2y = 12

5y = 9, so y = 1.8

Step 3: x = 1.8 + 1 = 2.8

Solution: x = 2.8, y = 1.8

3×3 System Example

Given System:

x + y + z = 6

2x - y + z = 3

x + 2y - z = 2

Using elimination:

• Eliminate z from equations 1 and 2

• Eliminate z from equations 1 and 3

• Solve the resulting 2×2 system

• Back-substitute to find all variables

Solution: x = 1, y = 2, z = 3

Real-World Applications

Economics

Supply and demand analysis, market equilibrium, cost optimization

Engineering

Circuit analysis, structural design, control systems

Physics

Force equilibrium, motion analysis, wave interference

Business

Resource allocation, production planning, financial modeling

Matrix Representation

Standard Form

Any system of linear equations can be written in matrix form as Ax = b, where:

  • A = coefficient matrix
  • x = variable vector
  • b = constant vector

Example:

[2 3] [x] [7]

[1 -1] [y] = [1]

Solution Methods

Cramer's Rule

Uses determinants to find solutions

Gaussian Elimination

Row operations to reach row echelon form

Matrix Inversion

x = A⁻¹b when A is invertible

Problem-Solving Tips

Before You Start

  • • Write equations in standard form
  • • Check for obvious solutions first
  • • Choose the most efficient method
  • • Look for patterns in coefficients

Common Pitfalls

  • • Sign errors during elimination
  • • Forgetting to check solutions
  • • Misaligning coefficients
  • • Dividing by zero