Calculator for Simultaneous Equations

Simultaneous Equations Solver

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

a₁x + b₁y = c₁

a₂x + b₂y = c₂

Equation 1:

Equation 2:

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Understanding Simultaneous Equations

Simultaneous equations, also known as a system of equations, are a set of two or more equations that share the same variables. The goal is to find the values of these variables that satisfy all equations in the system simultaneously. For a system of two linear equations with two variables (commonly 'x' and 'y'), we are essentially looking for the point where two lines intersect on a graph.

The General Form

A common way to represent a system of two linear equations is:

a₁x + b₁y = c₁

a₂x + b₂y = c₂

Where:

• a₁ and a₂ are the coefficients of the 'x' variable in the first and second equations, respectively.
• b₁ and b₂ are the coefficients of the 'y' variable in the first and second equations, respectively.
• c₁ and c₂ are the constant terms in the first and second equations, respectively.

Methods for Solving Simultaneous Equations

There are several methods to solve simultaneous equations:

1. Substitution Method: Solve one equation for one variable in terms of the other, then substitute that expression into the second equation.
2. Elimination Method: Multiply one or both equations by a constant so that one of the variables has coefficients that are opposites. Then, add the equations together to eliminate that variable.
3. Graphical Method: Graph both equations on the same coordinate plane. The point of intersection (if any) represents the solution.
4. 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.

How This Calculator Works (Cramer's Rule)

Our calculator utilizes Cramer's Rule to efficiently solve systems of two linear equations. Cramer's Rule involves calculating three determinants:

• Determinant of the Coefficient Matrix (D): This is calculated from the coefficients of x and y: D = (a₁ * b₂) - (a₂ * b₁)
• Determinant for x (Dₓ): This is found by replacing the x-coefficients in the coefficient matrix with the constant terms: Dₓ = (c₁ * b₂) - (c₂ * b₁)
• Determinant for y (Dᵧ): This is found by replacing the y-coefficients in the coefficient matrix with the constant terms: Dᵧ = (a₁ * c₂) - (a₂ * c₁)

Once these determinants are calculated, the solutions for x and y are given by:

x = Dₓ / D

y = Dᵧ / D

Interpreting the Results

• Unique Solution: If D ≠ 0, there is a unique solution for x and y, meaning the two lines intersect at a single point.
• No Solution: If D = 0 but Dₓ ≠ 0 or Dᵧ ≠ 0, the system has no solution. This indicates that the lines are parallel and distinct, never intersecting.
• Infinitely Many Solutions: If D = 0, Dₓ = 0, and Dᵧ = 0, the system has infinitely many solutions. This means the two equations represent the same line, and every point on that line is a solution.

Example Calculation

Let's solve the following system of equations:

Equation 1: 2x + 3y = 7

Equation 2: 4x - 2y = 6

Here, we have:

• a₁ = 2, b₁ = 3, c₁ = 7
• a₂ = 4, b₂ = -2, c₂ = 6

Using Cramer's Rule:

• D = (2 * -2) - (4 * 3) = -4 - 12 = -16
• Dₓ = (7 * -2) - (6 * 3) = -14 - 18 = -32
• Dᵧ = (2 * 6) - (4 * 7) = 12 - 28 = -16

Now, we find x and y:

• x = Dₓ / D = -32 / -16 = 2
• y = Dᵧ / D = -16 / -16 = 1

So, the unique solution to this system is x = 2 and y = 1. You can verify this by plugging these values back into the original equations.