System of 3 Equations Calculator

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System of 3 Equations Calculator

System of 3 Equations Calculator

Enter the coefficients (a, b, c, d) for each of the three linear equations:

Equation 1: \(a_1 x + b_1 y + c_1 z = d_1\)

Equation 2: \(a_2 x + b_2 y + c_2 z = d_2\)

Equation 3: \(a_3 x + b_3 y + c_3 z = d_3\)

Enter coefficients and click "Calculate Solution".

Understanding Systems of 3 Linear Equations

A system of three linear equations with three variables (commonly x, y, and z) represents three distinct planes in three-dimensional space. Finding a solution to such a system means finding the point (or points) where all three planes intersect.

Mathematical Methods

There are several common methods to solve these systems:

  • 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 using similar substitution or elimination methods.
  • Elimination (or Addition): Multiply equations by constants so that when you add or subtract equations, one variable cancels out. Repeat this process to reduce the system to a single variable.
  • Matrix Methods (Cramer's Rule or Gaussian Elimination): Represent the system as matrices and use matrix operations. Cramer's Rule uses determinants, while Gaussian Elimination uses row operations to transform the matrix into a simpler form (row echelon form or reduced row echelon form) from which the solution can be read.

Cramer's Rule Explained

For a system of the form:

\(a_1 x + b_1 y + c_1 z = d_1\) \(a_2 x + b_2 y + c_2 z = d_2\) \(a_3 x + b_3 y + c_3 z = d_3\)

We can define the following determinants:

The determinant of the coefficient matrix (D):

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

The determinant Dₓ is found by replacing the x-coefficients column with the constants column:

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

Similarly, for Dᵧ and D<0xE1><0xB5><0xA3>:

Dᵧ = | a₁ d₁ c₁ | D<0xE1><0xB5><0xA3> = | a₁ b₁ d₁ | | a₂ d₂ c₂ | | a₂ b₂ d₂ | | a₃ d₃ c₃ | | a₃ b₃ d₃ |

If \(D \neq 0\), the system has a unique solution given by:

x = Dₓ / D
y = Dᵧ / D
z = D<0xE1><0xB5><0xA3> / D

If \(D = 0\), the system either has no solution (inconsistent) or infinitely many solutions (dependent).

Use Cases

Systems of linear equations are fundamental in various fields:

  • Engineering: Analyzing circuits, structural loads, and control systems.
  • Economics: Modeling market equilibria, resource allocation, and input-output analysis.
  • Computer Graphics: Transformations and solving for positions or orientations.
  • Physics: Solving problems in mechanics, electromagnetism, and quantum mechanics.
  • Operations Research: Optimization problems and scheduling.

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