Gaussian Elimination Method

Calculator for systematic solution of linear systems with detailed step-by-step guidance

Gauss Elimination Calculator

What is calculated?

The Gaussian elimination method solves linear systems by systematically transforming the coefficient matrix into row echelon form. It determines values for all unknowns through successive elimination.

Linear system (2×2)
a₁x + b₁y = c₁
a₂x + b₂y = c₂
1st equation a₁x + b₁y = c₁
2nd equation a₂x + b₂y = c₂
Solution of the linear system
Variable x
Variable y
The Gauss method transforms the system into a solvable form

Gauss Elimination Info

Solution types

Possible outcomes:

Unique solution Infinite solutions No solution

Principle: Systematic elimination of variables using equivalent row operations until row echelon form is reached.

Quick examples
x + y = 3
2x - y = 0:
→ x = 1, y = 2
x + y = 1
2x + 2y = 2:
→ Infinite solutions
x + y = 1
x + y = 2:
→ No solution (contradiction)
Related methods

→ Cramer's rule
→ Matrix arithmetic
→ Determinants

Gauss algorithm

Step 1: Elimination phase
1. Leading coefficients ≠ 0
2. Eliminate below the diagonal
3. Equivalent row operations
4. Reach row echelon form
Step 2: Back substitution
1. From bottom to top
2. Insert known values
3. Compute unknowns
4. Obtain solution
Standard form (2×2 system)
\[\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix}\]
Augmented matrix
\[\left(\begin{array}{cc|c} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{array}\right)\]
Row echelon form
\[\left(\begin{array}{cc|c} a_1 & b_1 & c_1 \\ 0 & b_2' & c_2' \end{array}\right)\]
Back substitution
\[y = \frac{c_2'}{b_2'}, \quad x = \frac{c_1 - b_1y}{a_1}\]

Detailed worked example

Example: x + 2y = 3 and 4x + 5y = 6

Original system:

\[\begin{align} x + 2y &= 3 \quad (I)\\ 4x + 5y &= 6 \quad (II) \end{align}\]

Augmented matrix:

\[\left(\begin{array}{cc|c} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right)\]

Step 1: Eliminate x in equation (II)

New equation (II'): (II) - 4·(I)

\[4x + 5y - 4(x + 2y) = 6 - 4(3)\] \[4x + 5y - 4x - 8y = 6 - 12\] \[-3y = -6\]

Row echelon form:

\[\begin{align} x + 2y &= 3\\ -3y &= -6 \end{align}\]

Back substitution:

\[y = \frac{-6}{-3} = 2\] \[x = 3 - 2y = 3 - 4 = -1\]

Solution: x = -1, y = 2
Check: (-1) + 2(2) = 3 ✓ and 4(-1) + 5(2) = 6 ✓

Elementary row operations

Allowed operations (equivalent transformations)
Swap

Swap two rows

\[Z_i \leftrightarrow Z_j\]

Does not change solution set

Multiply

Multiply a row by a non-zero constant

\[Z_i \rightarrow k \cdot Z_i\]

k ≠ 0 required

Add

Add a multiple of one row to another

\[Z_i \rightarrow Z_i + k \cdot Z_j\]

Elimination step

Special cases

Unique solution

Condition:

\[\det\begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \end{pmatrix} \neq 0\]

Exactly one intersection of the lines

Infinite solutions

Condition:

\[\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\]

Identical lines

No solution

Condition:

\[\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}\]

Parallel lines

Practical applications

Engineering
  • Static calculations
  • Electrical networks
  • Material stresses
  • Optimization problems
Economics
  • Production planning
  • Cost optimization
  • Market equilibrium
  • Input-output models
Sciences
  • Chemical reaction equations
  • Physical models
  • Population dynamics
  • Signal processing

Historical context

Carl Friedrich Gauss (1777-1855)
  • "Princeps mathematicorum" - Prince of mathematicians
  • Systematization of the elimination method
  • Applications in geodesy and astronomy
  • Foundation for modern computational methods
Mathematical significance
  • Foundation of linear algebra
  • Basis for LU decomposition
  • Numerical methods in computer science
  • Complexity analysis O(n³)
Summary

The Gaussian elimination method is a fundamental algorithm of linear algebra that solves linear systems by systematic equivalent transformations. It forms the basis for many modern numerical methods and has broad applications in science, engineering and economics.

Mathematical Equations

Associative Property  •  Linear equation  •  Quadratic equation  •  Cubic equation  •  Discriminant  •  Gauss elimination method  •