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There are many instances in science and engineering where problem solving requires the simultaneous solution of a system of equations. Such a system of equations might look like the one in the figure below.

While there are many methods for solving sets of simultaneous equations, one of the better ways of doing so is to set-up an augmented matrix based on the coefficients of your variables and then to perform Gauss Elimination or Gauss-Jordan Elimination.

The process of Gauss elimination involves a series of matrix operations that reduce an augmented matrix into simpler forms from which the solution set of a system of equations can be more easily determined. Gauss elimination leaves a matrix in lower triangular form, with ones (1's) on the diagonal. From this point you can determine the final variable by inspection and then using back-substitution to determine the additional variables. For the case shown, you could immediately determine that x₃ = 10, and then solve the rows above to find x₂ and x₁.

Gauss-Jordan elimination continues the process further resulting in a diagonalized matrix in which the solution set appears in the final column. You could immediately tell from the figure above that x₁ = 5, x₂ = 10 and x₃ = 10.

There are three elementary row operations (ERO's) that may be performed during the process of Gauss elimination, without adversely affecting the solution. They are listed as follows:

Swap any two rows
Multiply any row by a constant
Add a multiple of one row to another row

In order to perform Gauss elimination most effectively, ERO's should be performed on a column-by-column basis, starting at the first column and ending at the second-to-last column. For each column the following systematic steps should be performed.

Get a one (1) on the diagonal element of the working column by multiplying the working row by the inverse of the diagonal element.
Get zeroes (0's) in each element of the working column above (for Gauss-Jordan) and below the diagonal element. This can be done by multiplying the diagonal element's row by the negative of the element in the row above or below the diagonal element.