# How do I convert this into a linear programming problem?

A farmer is planning to raise wheat and barley. Each acre of wheat yields a profit of \$50 and each acre of barley yields a profit of \$70. To sow the crop, two machines, a tractor and tiller, are rented. The tractor is available for 200 hours, and the tiller is available for 100 hours, Sowing an acre of barley requires 3 hours of tractor time and 2 hours of tilling. Sowing an acre of wheat requires 4 hours of tractor time and 1 hour of tilling. How many acres of each crop should be planted to maximize the farmer's profit?

I'll try to use simplex method, afterwards.

The first step is to describe what your variables are: $x$ which could be the number of acres of Barley sown, and $y$ which could be the number of acres of wheat.

Now, what is the profit function: $P = 70x + 50y$ (70 dollars per acre of barley + 50 dollars per acre of wheat).

Now, what are the constraints: certainly $x\geq 0$ and $y\geq 0$. What about not overusing the tiller?

$$2x + 1y \leq 100$$ 2 hours on tiller for each acre of barley, 1 hour for each acre of wheat... not more than 100 hours available.

Similarly, what about not overusing the tractor?

$$3x + 4y \leq 200$$ 3 hours on tractor for each acre of barley, 4 hours on the tractor for each acre of wheat... not more than 200 hours total on the tractor.

Condensing all those constraints you have:

Maximize $P=70x + 50y$

Subject to \begin{align*} 2x + 1y &\leq 100 \\ 3x + 4y &\leq 200 \\ x, y &\geq 0 \end{align*}