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# Gaussian Elimination

 **Gaussian elimination** is an algorithm to solve systems of equations.

Process to solve problems: 

{% 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

![](/files/-LeS-Uh3XigwxPb4R5El)

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 

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


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