Gaussian Elimination Calculator

1. Define Your System

Enter coefficients for [A] and constants for [B]. Use arrow keys to navigate.

2. Step-by-Step Solution

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Learn About Gaussian Elimination

Gaussian Elimination is a powerful and systematic algorithm used in linear algebra to solve systems of linear equations. The core idea is to simplify a complex system of equations into a much easier form (called row echelon form) from which the solutions can be found by simple substitution.

The Core Process

  1. Construct the Augmented Matrix: A system of equations is first converted into an "augmented matrix." This is a compact way to represent the system, where the coefficients of the variables form one part of the matrix (A) and the constants on the other side of the 'equals' sign form the final column (B). The matrix is written as [A|B].
  2. Forward Elimination (Creating Row Echelon Form): The main goal is to introduce zeros below the main diagonal of the coefficient part of the matrix. This is achieved by applying a sequence of Elementary Row Operations:
    • Swapping: Exchanging the positions of two rows.
    • Scaling: Multiplying an entire row by a non-zero number.
    • Replacement: Adding a multiple of one row to another row. (This is the key operation for creating zeros).
  3. Back Substitution (Finding the Solution): Once the matrix is in row echelon form, the last equation will have only one variable, making it easy to solve. You take this solution, substitute it into the second-to-last equation to find the next variable, and continue this process, working your way back to the top.

A Simple Example

Consider the system: x + 2y = 5 and 3x + 4y = 11. The augmented matrix is:

To eliminate the 3 in the second row, we perform the row operation R₂ → R₂ - 3*R₁. This gives us the row echelon form:

From the second row, we can now easily solve for y: -2y = -4, which means y = 2. Substituting this value back into the first equation (x + 2(2) = 5) gives us x + 4 = 5, so x = 1. The solution is (1, 2).