If you've gotten this far, you should know how to: Start at a point with a basis of variables that are 0, forming the pivot columns of a row reduced matrix. Determine whether exchanging one of those basis variables will increase or decrease the object function. Exchange a basis variable in order to increase the objective unction and put the equations into row reduced echelon form with respect to the new basis. The simplex algorithm is to continue in this manner, always performing basis exchanges which improve the objective function, until no more exchanges are possible. We conclude with an example: Buzz Buzz Buzz Coffee has on hand 1 kg of coffee grounds, 1 gallon of milk and 10 cups of sugar They can use these to make espressos, containing 8 grams of grounds and no milk or sugar: lattes, containing 15 grams of grounds, 0.0625 gallons of milk and 0.125 cups of sugar: or cafe cubano containing 7.5 grams of grounds, no milk and 0.125 cups of sugar. They will be able to sell all they produce, which they will sell at prices of $2 for espressos, $4 for lattes and $5 for a cafe cubano. Let e, l and c be the number of espressos, lattes and cafes cubanos manufactured, and let g, m and s be the amounts of grounds, milk and sugar left over when they are done. Let p be the amount of money they take in. Record the linear equations relating e, l, c, g, m, s and p. Start at the point where no drinks are made (so e = l = c = 0). Exchange one of these variables, in order to increase p. Repeat the process of exchanging a basis variable to increase p until there are no exchanges which will make p larger. How many of each drink should be made?