The first constraint is
2x + 2y + z 
8.
To turn it into an equation, we must add a slack variable s to the left-hand side, getting
The next constraint is
x + 4y 
3z 12,
and we add the slack variable t to the left-hand side, getting
x + 4y 3z + t = 12.
Finally, the objective is
We must subtract 12x + 15y + 5z from both sides to get the desired equation:
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