]> Exercise 32.2

## Exercise 32.2

Perform Gaussian elimination on the following set of equations to find a solution

$x + 3 y − z = 7 3 x + y − 2 z = 4 − x − y + z − 1$

Solution:

Here if you add the first and third equations you are lucky to find that the sum equation is $2 y = 8$ , and you deduce $y = 4$ . Adding the second and twice the third equation yields $x − y = 6$ , or $x = 10$ . Substituting in the third equation you find $− 10 − 4 + z = 1$ which tells you $z = 15$ , and you have your solution. Testing the solution in the original equations you get $10 + 12 − 15 = 7 , 30 + 4 − 30 = 4$ , and $− 10 − 4 + 15 = 1$ , which are all correct.