System of Linear Equations Calculator

System of Linear Equations Solver (2x2)

This calculator helps you solve systems of two linear equations with two variables (x and y). Input your equations in the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to get the values for x and y instantly.

Coefficient a₁ (Equation 1)

e.g., 2 for 2x

Coefficient b₁ (Equation 1)

e.g., 3 for 3y

Constant c₁ (Equation 1)

e.g., 7

Coefficient a₂ (Equation 2)

e.g., 4 for 4x

Coefficient b₂ (Equation 2)

e.g., -1 for -y

Constant c₂ (Equation 2)

e.g., 1

How it works

This calculator helps you solve systems of two linear equations with two variables (x and y). Input your equations in the form a₁x + b₁y = c₁ and a₂x + b₂y = c₂ to get the values for x and y instantly.

The Formula

For a 2x2 system:
a₁x + b₁y = c₁
a₂x + b₂y = c₂

Worked Example

Example: Solving a 2x2 System
Consider the system: 2x + 3y = 7 4x - y = 1 Input a1=2, b1=3, c1=7, a2=4, b2=-1, c2=1. The calculator will output x=5/7, y=13/7.

Tips, Assumptions & Limitations

Ensure all coefficients and constants are entered correctly.
Use fractions or decimals as needed (e.g., 1/2 as 0.5).
The calculator will indicate if there's no unique solution (parallel lines or identical lines).

FAQ

A system of linear equations is a set of two or more linear equations that share the same variables. The goal is to find values for these variables that simultaneously satisfy all equations in the system.

You can input negative numbers directly (e.g., -5). For fractions, convert them to decimals before entering (e.g., 1/2 becomes 0.5). The calculator will handle the arithmetic.

If the calculator indicates 'no solution', it means the two lines represented by your equations are parallel and distinct, so they never intersect. There are no (x, y) values that satisfy both equations.

This occurs when the two equations represent the exact same line. Any point on that line is a solution, meaning there are an infinite number of (x, y) pairs that satisfy both equations.