Solution to Gaussian Elimination

Given the set of linear equations:

Let's look at the solution graphically first.

Figure 1:

Figure 2:

Figure 3:

Figure 1 is the plane of , Figure 2 is the plane of , and Figure 3 is the plane of .

The final result will be the intersection of the three planes:

One way of solving the equations:

1. Divide by 2; multiply by 3 and subtract it from to cancel from ; subtract the new from to cancel from :

   

2. Divide by -18; multiply by 2 and add it to to cancel from :

   

3. Now, solve for , and substitute it into to solve for ; then substitute and into and solve for :

   

So, = -2, = 9 and = 3.

Solving the equations in a matrix:

Two main steps are involved in this solution. The Gaussian elimination is performed first, followed by the Back-substitution:

• Reducing the matrix with Gaussian elimination

  

• Back-substitution

  Reading from the matrix:
  

  • finding

    

  • finding

    

  • finding

    

So, = -2, = 9 and = 3. As expected, the two different methods give the same answer.