System of Equations Calculator

What Is a System of Equations

A system of equations is a set of two or more equations that share the same variables and must be solved together. In algebra, the most common case is a system of two linear equations with two unknowns:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

The goal is to find values of x and y that satisfy both equations at the same time.

How to Solve a System of Linear Equations

There are several methods used to solve systems of equations:

• Substitution method
• Elimination (addition) method
• Cramer's rule
• Graphical method

This calculator uses Cramer's rule, which is a fast and reliable way to solve a system of two linear equations.

Cramer's Rule Formula

For the system:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

First calculate the main determinant:

D = a₁b₂ − a₂b₁

Then calculate the auxiliary determinants:

Dx = c₁b₂ − c₂b₁
Dy = a₁c₂ − a₂c₁

Finally compute the solutions:

x = Dx / D
y = Dy / D

How Many Solutions Can a System Have

Depending on the value of the determinant, a system of equations can have different types of solutions:

• D ≠ 0 — the system has exactly one solution
• D = 0 and Dx = 0, Dy = 0 — the system has infinitely many solutions
• D = 0 but Dx ≠ 0 or Dy ≠ 0 — the system has no solution

Example of Solving a System of Equations

Consider the system:

2x + 3y = 13
x − y = 1

Calculate the determinants:

D = 2 · (-1) − 1 · 3 = -5
Dx = 13 · (-1) − 1 · 3 = -16
Dy = 2 · 1 − 13 · 1 = -11

Then compute:

x = -16 / -5 = 3.2
y = -11 / -5 = 2.2

These are exactly the values calculated by the CalcCore system of equations calculator.

Where Systems of Equations Are Used

Systems of equations are widely used in mathematics, engineering, physics, economics, and computer science. They allow us to model situations where multiple variables interact with each other.

Common applications include:

• physics calculations
• engineering problems
• economic models
• data analysis
• optimization problems