3 System of Equations Calculator

Equation Coefficients

Enter the coefficients for the following system of linear equations:

a1*x + b1*y + c1*z = d1
a2*x + b2*y + c2*z = d2
a3*x + b3*y + c3*z = d3

Understanding Systems of Linear Equations

A system of linear equations is a collection of two or more linear equations that involve the same set of variables. In this calculator, we focus on a system of three linear equations with three variables (commonly denoted as x, y, and z). The goal is to find a unique set of values for x, y, and z that satisfies all equations simultaneously.

A system can be represented in the general form:

• a₁x + b₁y + c₁z = d₁
• a₂x + b₂y + c₂z = d₂
• a₃x + b₃y + c₃z = d₃

Here, 'a', 'b', and 'c' are the coefficients of the variables, and 'd' is the constant term on the right-hand side of each equation.

Methods for Solving

Several methods can be used to solve such systems, including:

• Substitution Method: Solve one equation for one variable and substitute that expression into the other equations.
• Elimination Method: Multiply equations by constants so that when two equations are added or subtracted, one variable is eliminated. This process is repeated to reduce the system to a single equation with one variable.
• Matrix Methods (e.g., Cramer's Rule, Gaussian Elimination): These methods involve representing the system in matrix form and using matrix operations to find the solution. Cramer's Rule is particularly useful for systems where the number of equations equals the number of variables and the determinant of the coefficient matrix is non-zero.

How this Calculator Works (Cramer's Rule)

This calculator uses Cramer's Rule, a method that relies on determinants. For a system to have a unique solution using Cramer's Rule, the determinant of the coefficient matrix (D) must be non-zero.

The determinant of the coefficient matrix is:

    D = | a1 b1 c1 |
        | a2 b2 c2 |
        | a3 b3 c3 |
        

To find the values of x, y, and z, we calculate three additional determinants:

• Dx: Replace the x-coefficients (a1, a2, a3) with the constants (d1, d2, d3).

        Dx = | d1 b1 c1 |
             | d2 b2 c2 |
             | d3 b3 c3 |
            

• Dy: Replace the y-coefficients (b1, b2, b3) with the constants (d1, d2, d3).

        Dy = | a1 d1 c1 |
             | a2 d2 c2 |
             | a3 d3 c3 |
            

• Dz: Replace the z-coefficients (c1, c2, c3) with the constants (d1, d2, d3).

        Dz = | a1 b1 d1 |
             | a2 b2 d2 |
             | a3 b3 d3 |
            

The solution is then given by:

• x = Dx / D
• y = Dy / D
• z = Dz / D

If D = 0, the system either has no solution (inconsistent) or infinitely many solutions (dependent), and Cramer's Rule cannot be directly applied to find a unique solution. This calculator will indicate such cases.

Use Cases

Systems of linear equations are fundamental in many fields:

• Engineering: Analyzing circuits, structures, and control systems.
• Physics: Modeling physical phenomena, from mechanics to electromagnetism.
• Economics: Balancing supply and demand, and modeling market equilibrium.
• Computer Graphics: Transformations and rendering.
• Operations Research: Optimization problems.

This calculator provides a quick and accurate way to solve such systems, assisting students, researchers, and professionals in verifying solutions or exploring different scenarios.