Let us say that you are sitting at a coffee shop and you want to impress your significant other by using your linear algebra skills.

Suppose that you want to determine the strength of the caf au lait you're drinking. You know your caf au lait has a certain amount of coffee (x), it is brewed to achieve a certain strength (Sc), and it has a specific amount of milk (y) that could be whole, fat-free, or some other type (Sm). So to determine the strength of your drink, you need to know the quantity and the strength of both ingredients.

You set your caf au lait equation as follows:

x coffee Sc + y milk Sm = z (coffee S'c milk S'm)

where

x is the quantity of coffee with a known strength Sc

y the quantity of milk with a known strength Sm

z is the volume of a cup of caf au lait with the combined strength (S'c and S'm).

This equation looks almost chemical, such as:

x Hydrogen + y Oxygen = z H2O (water)

1. Describe the caf au lait equation above. 2. Choose values of Sc, Sm and S'c, S'm as described below: Sc = 2, Sm = 4 and S'c = 1, S'm = 2 3. Describe how to set up the system of linear equations to solve for x and y. 4. Determine the matrices involved in solving this system of equations. 5. Explain how to use Cramer's rule and find the values of x and y as a function of z.