How did you know there was only 1 solution if you didn't know how to do
the other part?  Did you just guess?



That is the matrix for this system of equations:



Solve the bottom equation for P:



We want P to be as large as possible, and we have three non-negative numbers
subtracted from the 78.  We can keep the whole 78 for P by choosing all three
variables z, v, and w = 0.  So we substitute those in the system and get:



So P has a maximum value of 78 when x = 15, y = 3, and z = 0.  The point is

(x,y,z) = (15,3,0)

Edwin