In function P(x,y) = 85x – (50y + 150000). What value indicates the increase…

Concept: For a multivariable function, holding one variable constant turns the expression into a linear function of the remaining variable alone; the rate of increase in that function per unit increase in the free variable is simply the coefficient of that variable — the partial derivative with respect to it, since every other term (constants and terms in the held-constant variable) drops out under differentiation.

Application:

1. Write P(x, y) = 85x – (50y + 150000) = 85x – 50y – 150000.

2. Hold y constant, so both –50y and –150000 are fixed numbers that do not change as x varies.

3. This reduces P to a function of x alone of the form P(x) = 85x + k, where k = –50y – 150000 is a constant.

4. The rate of increase in P per unit increase in x is the coefficient of x in this expression, i.e. 85 (equivalently, ∂P/∂x = 85).

Cross-check: Fixing y at any value, say y = 0, gives P(0,0) = –150000 and P(1,0) = 85 – 150000; the difference P(1,0) – P(0,0) = 85, confirming the coefficient of x is exactly the increase in P for each unit increase in x.

Answer: The value 85 is correct.