Two different products, P$1$ and P$2,$ can be manufactured by one or both of two different machines, M$1$ and M$2.$ The unit processing time of either product on either machine is the same. The daily capacity of machine M$1$ is $200$ units $($of either P$1$ or P$2,$ or a mix of both$),$ and the daily capacity of machine M$2$ is $250$ units. The shop supervisor wants to balance the production schedule of the two machines such that the total number of units produced on one machine is within $5$ units of the number produced on the other. The profit per unit of P$1$ is $$10$ and that of P$2$ is $$15.$ Set up the problem as an LP in equation form. Find the optimal solution