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An important thing to pay attention to while solving for this problem is all of the signs and to make multiply correctly each row. It is extremely important to add or subtract the right rows when finding a certain zero because if not that automatically get you on the wrong track. There are a lot of little steps that you must do correct. Using the calculator after solving your matrix is helpful to see if your answer is correct.
* I did not verbally explain how to solve for x at the end but you simply plug in your z and y which were already solved for.