$2.$ You are a CEO!

You are building a company centered around a cool design for a T$-$shirt.

Step $1$ The Design $-$ production costs and selling price

$-$ Let's look at the production costs for the T$-$shirts. No matter how many T$-$shirts you produce, you pay the factory $\backslash (\backslash$$ $90\backslash )$ for electricity. Then, you pay $\backslash (\backslash$$ $7\backslash )$ per T$-$shirts to make it$.$ You want to sell the shirts for $\backslash (\backslash$$ $20\backslash )$ each. Write the equation for the production costs Pt$.$

$-$ Write the equation for the selling revenue from the T$-$shirts, St$.$

Step $2$ Diversification

$-$ Obviously, you cannot have a profitable company selling just T$-$shirts. So you decide to put your design on other products to increase the rentability of the whole project. In addition to T$-$shirts, you will sell mugs and keychains.

$-$ You can sell mugs for $\backslash (\backslash$$ $13\backslash )$ and they cost $\backslash (\backslash$$ $10\backslash )$ to make.

$-$ Keychains are cheap to make, they cost $\backslash (\backslash$$ $2\backslash )$ per keychain plus $\backslash (\backslash$$ $2\backslash )$ of fixed cost.

$-$ Keychains can be sold for $\backslash (\backslash$$ $5\backslash ).$

$-$ Write the equations for the production costs and selling revenues of the mugs $\backslash (\backslash$mathbf$\{$P m$\},\backslash$mathbf$\{$S m$\}\backslash )$ and the keychains $\backslash (\backslash$mathbf$\{$P k$\},\backslash$mathbf$\{$S k$\}\backslash ).$

Step $3$ How much can you sell?

$-$ Realistically, it will be much easier to sell keychains on campus, so you expect to sell $10$ times the amount of T$-$shirts.

$-$ You figured out researchers drink a lot of coffee so you show up at every conference on campus and sell a large quantity of mugs: half the amount of keychains plus twice the amount of T$-$shirts.

$-$ Write the equations for the number of mugs $\backslash (\backslash$mathbf$\{$N m$\}\backslash )$ and keychains $\backslash (\backslash$mathbf$\{$N k$\}\backslash ).$