![extended gaussian image from vector code extended gaussian image from vector code](https://thumbs.dreamstime.com/b/abstract-background-futuristic-technology-bubble-glowing-cryptocurrency-dai-stable-coin-222546897.jpg)
This is called the augmented matrix, and each row corresponds to an equation in the given system. Next, the coefficient matrix is augmented by writing the constants that appear on the right‐hand sides of the equations as an additional column:
![extended gaussian image from vector code extended gaussian image from vector code](http://codes.arizona.edu/toolbox/help/html/svm_eq13198901335625790277.png)
This is called the coefficient matrix of the system. The first step is to write the coefficients of the unknowns in a matrix: The previous example will be redone using matrices. This method reduces the effort in finding the solutions by eliminating the need to explicitly write the variables at each step. Gaussian elimination is usually carried out using matrices. (Back‐substitution of y = 1 into the original second equation, 3 x − 2 y = 4, would also yeild x = 2.) The solution of this system is therefore ( x, y) = (2, 1), as noted in Example 1. Back‐substitution of y = 1 into the original first equation, x + y = 3, yields x = 2. This final equation, −5 y = −5, immediately implies y = 1. Multiplying the first equation by −3 and adding the result to the second equation eliminates the variable x: This method, characterized by step‐by‐step elimination of the variables, is called Gaussian elimination. Once this final variable is determined, its value is substituted back into the other equations in order to evaluate the remaining unknowns. The fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. The purpose of this article is to describe how the solutions to a linear system are actually found.