# Example of Solving a System of Linear Equations by Gaussian Elimination.

### This solution has been done by the calculator presented on the site.

Please note that the coefficients will disappear which located in the "red" positions.

- | 4 | x_{1} | + | 5 | x_{2} | - | 3 | x_{3} | + | 3 | x_{4} | = | 20 | |

4 | x_{1} | + | 2 | x_{2} | + | 3 | x_{3} | + | 4 | x_{4} | = | 10 | ||

5 | x_{1} | + | 4 | x_{2} | + | 4 | x_{3} | + | 3 | x_{4} | = | 20 |

The equation 3 is added to the equation 1. more info

( -4 x

_{1}+ 5 x_{1}) + ( 5 x

_{2}+ 4 x_{2}) + ( -3 x

_{3}+ 4 x_{3}) + ( 3 x

_{4}+ 3 x_{4}) = 20 + 20

This transformation will allow us to count without fractions for some time.

x_{1} | + | 9 | x_{2} | + | x_{3} | + | 6 | x_{4} | = | 40 | ||||

4 | x_{1} | + | 2 | x_{2} | + | 3 | x_{3} | + | 4 | x_{4} | = | 10 | ||

5 | x_{1} | + | 4 | x_{2} | + | 4 | x_{3} | + | 3 | x_{4} | = | 20 |

The equation 1 multiplied by -4 is added to the equation 2. more info

( 4 x

_{1}+ x_{1}* ( -4) ) + ( 2 x

_{2}+ 9 x_{2}* ( -4) ) + ( 3 x

_{3}+ x_{3}* ( -4) ) + ( 4 x

_{4}+ 6 x_{4}* ( -4) ) = 10 + 40 * ( -4)

The "red" coefficient is zero.

x_{1} | + | 9 | x_{2} | + | x_{3} | + | 6 | x_{4} | = | 40 | ||||

- | 34 | x_{2} | - | x_{3} | - | 20 | x_{4} | = | - 150 | |||||

5 | x_{1} | + | 4 | x_{2} | + | 4 | x_{3} | + | 3 | x_{4} | = | 20 |

The equation 1 multiplied by -5 is added to the equation 3. more info

( 5 x

_{1}+ x_{1}* ( -5) ) + ( 4 x

_{2}+ 9 x_{2}* ( -5) ) + ( 4 x

_{3}+ x_{3}* ( -5) ) + ( 3 x

_{4}+ 6 x_{4}* ( -5) ) = 20 + 40 * ( -5)

The "red" coefficient is zero.

x_{1} | + | 9 | x_{2} | + | x_{3} | + | 6 | x_{4} | = | 40 | ||||

- | 34 | x_{2} | - | x_{3} | - | 20 | x_{4} | = | - 150 | |||||

- | 41 | x_{2} | - | x_{3} | - | 27 | x_{4} | = | - 180 |

The equation 2 multiplied by -41/34 is added to the equation 3. more info

( -41 x

_{2}+ ( -34 x_{2}) * ( -41/34) ) + ( - x

_{3}+ ( - x_{3}) * ( -41/34) ) + ( -27 x

_{4}+ ( -20 x_{4}) * ( -41/34) ) = -180 + ( -150) * ( -41/34)

The "red" coefficient is zero.

x_{1} | + | 9 | x_{2} | + | x_{3} | + | 6 | x_{4} | = | 40 | ||||

- | 34 | x_{2} | - | x_{3} | - | 20 | x_{4} | = | - 150 | |||||

7/34 | x_{3} | - | 49/17 | x_{4} | = | 15/17 |

We will find the variable x

_{3}from equation 3 of the system. 7/34 x

_{3}- 49/17 x_{4}= 15/17x

_{3}= 30/7 + 14 x_{4} We will find the variable x

_{2}from equation 2 of the system. - 34 x

_{2}- x_{3}- 20 x_{4}= - 150- 34 x

_{2}= - 150 + x_{3}+ 20 x_{4}- 34 x

_{2}= - 150 + ( 30/7 + 14 x_{4}) + 20 x_{4}x

_{2}= 30/7 - x_{4} We will find the variable x

_{1}from equation 1 of the system. x

_{1}+ 9 x_{2}+ x_{3}+ 6 x_{4}= 40 x

_{1}= 40 - 9 x_{2}- x_{3}- 6 x_{4} x

_{1}= 40 - 9 * ( 30/7 - x_{4}) - ( 30/7 + 14 x_{4}) - 6 x_{4}x

_{1}= - 20/7 - 11 x_{4}Result:

x

_{1}= - 20/7 - 11 x_{4}x

_{2}= 30/7 - x_{4}x

_{3}= 30/7 + 14 x_{4}