# Gaussian Elimination

&#x20;**Gaussian elimination** is an algorithm to solve systems of equations.

Process to solve problems:&#x20;

{% hint style="info" %}
* Determine a “pivot,” i.e. the top leftmost nonzero number that isn’t in a row/column already used. If necessary, scale this whole row to make the pivot entry 1.
* For each row  below the pivot row  multiply the pivot row by the leading entry and subtract:  Each row below the pivot row will be left with zeros in that column.
* Repeat until there are no more pivots. At this point, the matrix will have all 0s in the lower left triangle.
{% endhint %}

For example, starting with the above matrix, we have

![](https://846345873-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-LagOeJ2nL90MQERwhxy%2F-LeRWV75IJvPXtiaJVGM%2F-LeS-Uh3XigwxPb4R5El%2Fimage.png?alt=media\&token=941577c6-58bb-4edd-a72f-7f47d1ffd8f4)

Here, the pivots are shown in blue. Now let’s look at that last row--the equation it represents is z = 1. Then we can find and directly.

**Gaussian elimination** provides solutions to matrix equations of the form **Ax = b** where **A** is the matrix of&#x20;

coefficients, **x** is the matrix of variables, **b** and is the matrix of RHSs (right hand side).
