Solve System of Equations Calculator

System of Linear Equations Solver (2×2)

Enter the coefficients and constants for your two linear equations in the form:

Equation 1: a₁x + b₁y = c₁

Equation 2: a₂x + b₂y = c₂

Coefficient of x in Eq 1 (a₁):

Coefficient of y in Eq 1 (b₁):

Constant in Eq 1 (c₁):

Coefficient of x in Eq 2 (a₂):

Coefficient of y in Eq 2 (b₂):

Constant in Eq 2 (c₂):

Solution:

Understanding and Solving Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations involving the same set of variables. When we talk about "solving" a system, we are looking for the values of the variables that satisfy all equations in the system simultaneously. For a system of two linear equations with two variables (commonly x and y), this solution represents the point where the two lines intersect on a graph.

The General Form of a 2×2 System

A system of two linear equations with two variables can be written in the general form:

a₁x + b₁y = c₁ (Equation 1)

a₂x + b₂y = c₂ (Equation 2)

Here, a₁, b₁, c₁, a₂, b₂, and c₂ are coefficients and constants, which are real numbers. The variables are x and y.

Methods for Solving Systems of Equations

There are several common methods to solve systems of linear equations:

Substitution Method: Solve one equation for one variable, then substitute that expression into the other equation.
Elimination Method (Addition Method): Multiply one or both equations by a constant so that the coefficients of one variable are opposites, then add the equations together to eliminate that variable.
Graphical Method: Graph both lines on the same coordinate plane. The point of intersection is the solution. This method can be less precise for non-integer solutions.
Matrix Method (Cramer's Rule): This algebraic method uses determinants to find the values of the variables. It's particularly efficient for 2×2 and 3×3 systems. Our calculator above uses a form of Cramer's Rule.

Types of Solutions

When solving a system of two linear equations, there are three possible outcomes:

Unique Solution: The most common case, where the two lines intersect at exactly one point. This means there is one specific value for x and one specific value for y that satisfies both equations.
Example: x = 2, y = 3
No Solution (Inconsistent System): The two lines are parallel and never intersect. This occurs when the slopes of the lines are the same, but their y-intercepts are different. Algebraically, you might end up with a false statement like 0 = 5.
Example: x + y = 5 and x + y = 10
Infinitely Many Solutions (Dependent System): The two equations represent the exact same line. This means every point on the line is a solution, and there are an infinite number of such points. Algebraically, you might end up with a true statement like 0 = 0.
Example: 2x + 2y = 10 and x + y = 5

How Our Calculator Works (Cramer's Rule for 2×2 Systems)

Our calculator uses Cramer's Rule, which is based on determinants. For a system:

a₁x + b₁y = c₁

a₂x + b₂y = c₂

We calculate three determinants:

D (Determinant of the coefficient matrix): D = a₁b₂ - a₂b₁
Dx (Determinant for x): Replace the x-coefficients with the constants: Dx = c₁b₂ - c₂b₁
Dy (Determinant for y): Replace the y-coefficients with the constants: Dy = a₁c₂ - a₂c₁

The solutions are then found by:

x = Dx / D
y = Dy / D

If D = 0, the system either has no solution or infinitely many solutions, as explained above.

Example Using the Calculator

Let's solve the following system:

Equation 1: 2x + 3y = 7

Equation 2: 4x - 1y = 1

Here are the values you would input into the calculator:

a₁ = 2
b₁ = 3
c₁ = 7
a₂ = 4
b₂ = -1
c₂ = 1

After clicking "Solve System", the calculator will output:

Solution for x: 0.7143 (or 5/7)
Solution for y: 1.8571 (or 13/7)

This means the lines intersect at the point (5/7, 13/7).