SOLUTION: Using the Gauss-Jordan elimination method, solve the following linear system. 7x +5y -3z = 16 3x -5y +2z = -8 5x +3y -7z = 0 x = y = z =

In the Gauss-Jordan elimination, you start with a system
of three equations and three unknowns:

  

Then you convert it to this matrix:



Then you use row operations and end up with a matrix like this:



That is, you get 0's in the lower left three elements.

Then you convert back to a system of equations:

   

Then you do what is called "back-substitution":

1. Solve the bottom equation for z.
2. Substitute that value of z in the middle equation, 
   and solve for y.
3. Substitute the values of y and z in the top equation
   and solve for x.  

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To get a 0 where the 3 is on the middle row:

Multiply the top row by -3 and the middle row by 7, and
add them together:


--------------


Replace the second row by that, leaving the rest as is



To get a 0 where the 5 is on the bottom row:

Multiply the top row by -5 and the bottom row by 7, and
add them together:


--------------


Notice that as it turns out, we can divide that through
by -2, so we might as well do that too, and get:



We replace the bottom row by that, leaving the rest as is



To get a 0 where the 2 is on the bottom row:

Take the middle row as it is. Multiply the bottom row by 25, and
add them together:


--------------


Notice that as it turns out, we can divide that through
by 448, so we might as well do that too, and get:



We replace the bottom row by that, leaving the rest as is



Then we convert that back to this system of equations:



or rather,



Now we do what is called "back-substitution":

The bottom equation is already solved for z.

Substitute  in the middle equation:







Substitute  and  in the top
equation:








So the solution is  

Edwin