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

Solution:

Here if you add the first and third equations you are lucky to find that the sum equation is 2y = 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.