You manage an ice cream factory that makes two flavors: Creamy Vanilla and Continental Mocha. Into each quart of Creamy Vanilla go 2 eggs and 3 cups of cream. Into each quart of Continental Mocha go 1 egg and 3 cups of cream. You have in stock 500 eggs and 900 cups of cream. You make a profit of $3 on each quart of Creamy Vanilla and $2 on each quart of Continental Mocha. How many quarts of each flavor should you make to earn the largest profit?

I am going to call x the number of quarts of Creamy Vanilla and y the number of quarts of Continental Mocha. The largest profit is going to be earned when all the items in stock are used.

The problem states that each quart of Creamy Vanilla uses 2 eggs and each quart of Continental Mocha uses 1 egg. There are 500 eggs in stock, so:

[tex]2x + y = 500[/tex]

The also problem states that each quart of Creamy Vanilla uses 3 cups of cream and each quart of Continental Mocha uses 3 cups of cream. There are 900 cups of cream in stock, so:

[tex]3x + 3y = 900[/tex]

We can simplify by 3, so:

[tex]x + y = 300[/tex]

Now we have to solve the following system of equations:

[tex]2x + y = 500[/tex]

[tex]x + y = 300[/tex]

I am going to multiply the second equation by -1 and add to the first, so we can find x

[tex]-x -y = -300[/tex]

.....

[tex]2x - x + y - y = 500 - 300[/tex]

[tex]x = 200[/tex]

You should make 200 quarts of Creamy Vanilla.

[tex]x + y = 300[/tex]

[tex]200 + y = 300[/tex]

[tex]y = 100[/tex]

You should make 100 quarts of Continental Mocha.