the mistake was when she multiplied the 2nd equation by 5, she forgot to also multiply that last number 10 by 5 as well.
correction :
5x + 3y - z = 15
-x + 2y + 3z = 10...multiply by 5
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5x + 3y - z = 15
-5x + 10y + 15z = 50 (result of multiplying by 5)
--------------------add
13y + 14z = 65
