System of Three Equations Calculator

Solve for x, y, and z in a system of three linear equations.

Equation 1

Equation 2

Equation 3

Understanding Systems of Three Linear Equations

A system of three linear equations with three variables (commonly denoted as x, y, and z) is a set of three distinct linear equations where each equation involves these three variables raised to the power of one. The goal is to find a unique set of values for x, y, and z that simultaneously satisfies all three equations.

A typical system looks like this:

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

Where a₁, b₁, c₁, d₁, a₂, b₂, c₂, d₂, a₃, b₃, c₃, and d₃ are all known coefficients (constants).

Methods for Solving

There are several algebraic methods to solve such systems, including:

• Substitution: Solve one equation for one variable and substitute that expression into the other two equations. This reduces the system to two equations with two variables, which can then be solved.
• Elimination (or Addition): Multiply one or more equations by constants so that the coefficients of one variable are opposites in two equations. Adding these equations eliminates that variable, reducing the system. Repeat for another variable to solve.
• Matrix Methods (e.g., Cramer's Rule or Gaussian Elimination): Represent the system as a matrix equation and use matrix operations to find the solution. This is particularly efficient for larger systems and is the basis for many computational solvers.

Cramer's Rule: The Mathematical Foundation

This calculator uses Cramer's Rule, a method that relies on determinants. For a system in the form:

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

We define the following determinants:

• Determinant of the coefficient matrix (D):

D = | a₁ b₁ c₁ | = a₁(b₂c₃ - b₃c₂) - b₁(a₂c₃ - a₃c₂) + c₁(a₂b₃ - a₃b₂)

| a₂ b₂ c₂ |

| a₃ b₃ c₃ |

• Determinant for x (Dx): Replace the x-coefficient column with the constants column (d₁, d₂, d₃).

Dx = | d₁ b₁ c₁ | = d₁(b₂c₃ - b₃c₂) - b₁(d₂c₃ - d₃c₂) + c₁(d₂b₃ - d₃b₂)

| d₂ b₂ c₂ |

| d₃ b₃ c₃ |

• Determinant for y (Dy): Replace the y-coefficient column with the constants column.

Dy = | a₁ d₁ c₁ | = a₁(d₂c₃ - d₃c₂) - d₁(a₂c₃ - a₃c₂) + c₁(a₂d₃ - a₃d₂)

| a₂ d₂ c₂ |

| a₃ d₃ c₃ |

• Determinant for z (Dz): Replace the z-coefficient column with the constants column.

Dz = | a₁ b₁ d₁ | = a₁(b₂d₃ - b₃d₂) - b₁(a₂d₃ - a₃d₂) + d₁(a₂b₃ - a₃b₂)

| a₂ b₂ d₂ |

| a₃ b₃ d₃ |

The solutions are then:

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

Special Cases:

• If D = 0, the system either has no unique solution (infinitely many solutions) or no solution at all. In such cases, Cramer's Rule is indeterminate.

Use Cases:

Systems of three linear equations are fundamental in many fields:

• Physics: Analyzing circuits, mechanics, and kinematics.
• Engineering: Solving structural analysis problems, fluid dynamics, and control systems.
• Economics: Modeling supply and demand, input-output analysis.
• Computer Graphics: Transformations and projections.
• Mathematics: As a foundational concept in linear algebra.

This calculator provides a quick and accurate way to solve these systems, helping students and professionals alike verify their work or find solutions efficiently.