Solving Systems of Equations Calculator

System of Linear Equations Solver (2×2)

Enter the coefficients for your system of two linear equations in the form:

a₁x + b₁y = c₁

a₂x + b₂y = c₂

Understanding 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 dealing with two variables, typically 'x' and 'y', a system usually consists of two equations. The goal is to find the values of 'x' and 'y' that satisfy all equations in the system simultaneously.

The General Form

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

• a₁x + b₁y = c₁
• a₂x + b₂y = c₂

Where a₁, b₁, c₁, a₂, b₂, c₂ are coefficients and constants, and x, y are the variables we aim to solve for.

Types of Solutions

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

1. Unique Solution: The lines represented by the two equations intersect at exactly one point. This means there is one specific pair of (x, y) values that satisfies both equations. This is the most common scenario.
2. No Solution (Inconsistent System): The lines represented by the two equations are parallel and never intersect. In this case, there are no (x, y) values that can satisfy both equations simultaneously. This occurs when the slopes are the same but the y-intercepts are different.
3. Infinitely Many Solutions (Dependent System): The two equations represent the exact same line. This means every point on the line is a solution, and thus there are an infinite number of (x, y) pairs that satisfy both equations. This happens when both the slopes and y-intercepts are identical.

Methods for Solving

Several methods can be used to solve systems of linear equations:

• Substitution Method: Solve one equation for one variable (e.g., solve for 'x' in terms of 'y'), then substitute that expression into the other equation.
• Elimination Method (Addition Method): Multiply one or both equations by constants so that when the equations are added or subtracted, one of the variables is eliminated.
• Cramer's Rule: This method uses determinants to find the values of the variables. It's particularly efficient for 2×2 and 3×3 systems and is often used in computational solvers.
• Graphical Method: Graph both equations on the same coordinate plane. The point of intersection (if any) is the solution.

How to Use This Calculator

This calculator uses Cramer's Rule to quickly find the solution to a system of two linear equations. Follow these steps:

1. Identify Coefficients: For each of your two equations, identify the coefficient of 'x' (a₁ and a₂), the coefficient of 'y' (b₁ and b₂), and the constant term (c₁ and c₂).
2. Input Values: Enter these numerical values into the corresponding input fields in the calculator.
3. Calculate: Click the "Solve System" button.
4. View Results: The calculator will display the values for 'x' and 'y' if a unique solution exists, or indicate if there are no solutions or infinitely many solutions.

Example Calculation

Let's solve the following system:

Equation 1: 2x + 3y = 7

Equation 2: 4x - 2y = 2

Here are the inputs for the calculator:

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

Using Cramer's Rule:

• Calculate the determinant D = a₁b₂ - a₂b₁ = (2)(-2) - (4)(3) = -4 - 12 = -16
• Calculate Dx = c₁b₂ - c₂b₁ = (7)(-2) - (2)(3) = -14 - 6 = -20
• Calculate Dy = a₁c₂ - a₂c₁ = (2)(2) - (4)(7) = 4 - 28 = -24

Since D ≠ 0, a unique solution exists:

• x = Dx / D = -20 / -16 = 1.25
• y = Dy / D = -24 / -16 = 1.5

The calculator would output: x = 1.25 and y = 1.5.