Gaussian elimination

Gaussian elimination, sometimes called simpy elimination, is a method in mathematics that is used to solve a system of linear equations. Such sets of equations occur throughout mathematics, physics, and even in the optimization of business practices, such as scheduling of bus routes, airlines, trains, and optimization of profits as a function of supplies and sales. The method can be accomplished using written equations, but is more often simplified using matrix forms of the equations.

Three basic manuevers are allowed in the Gaussian elimination method:

1. Interchanging any two equations
2. Multiplying both sides of any equation by a non-zero number
3. Adding a multiple of one equation to another equation

Consider the following system of linear equations that must all be satisfied simultaneously:

• (Eq. 1) $1X+1Y+2Z=0$
• (Eq. 2) $3X+1Y+2Z=2$
• (Eq. 3) $2X-2Y+1Z=-6$

Typically, the equation with the simplest form of X is placed on the top of the equation list. Subsequently, variable X is removed from the second or third equation by the addition or substract of the first equation, or a multiple thereof. Subtracting 3 times equation 1 from equation 2 eliminates X from equation 2, as shown below.

• (Eq. 1) $1X+1Y+2Z=0$
• (Eq. 2) $0X-2Y-4Z=2$
• (Eq. 3) $2X-2Y+1Z=-6$

After removing X from the third equation in a similar fashion, that is subtracting 2 times Eq. 1 from Eq. 2, one obtains

• (Eq. 1) $1X+1Y+2Z=0$
• (Eq. 2) $0X-2Y-4Z=2$
• (Eq. 3) $0X-4Y-1Z=-6$

Once one variable has been removed from all but one equation, and second variable is removed from all but one equation. In this set of equations for example,