# What is Gaussian elimination?

Jul 6, 2018

See below

#### Explanation:

Given: Gaussian elimination

Gaussian elimination, also known as row-reduction, is a technique used to solve systems of linear equations. The coefficients of the equations, including the constant are put in a matrix form.

Three types of operations are performed to create a matrix that has a diagonal of $1$ and $0 ' s$ underneath:

[ (1, a, b, c), (0, 1, d, e), (0, 0, 1, f) ]

The three operations are:

1. swap two rows
2. Multiply a row by a nonzero constant (scalar)
3. Multiply a row by a nonzero number and add to another row

Simple example. Solve for $x , y$ using Gaussian Elimination:

$2 x + 4 y = - 14$
$5 x - 2 y = 10$

Becomes:
[ (2, 4, -14), (5, -2, 10) ]

Multiply row 1 by $\frac{1}{2}$:
[ (1, 2, -7), (5, -2, 10) ]

Replace row 2 with: Multiply row 1 by $- 5$ and add to row 2:
[ (1, 2, -7), (0, -12, 45) ]

Divide row 2 by $- 12$:
[ (1, 2, -7), (0, 1, -15/4) ] => x + 2y = -7; " "y = -15/4

Use back substitution to solve for $x$ and $y$:

$x + \frac{2}{1} \left(- \frac{15}{4}\right) = - 7$

$x - \frac{30}{4} = - 7$

$x - \frac{15}{2} = - \frac{14}{2}$

$x = - \frac{14}{2} + \frac{15}{2} = \frac{1}{2}$

Solution: $\left(\frac{1}{2} , - \frac{15}{4}\right)$