1 EXPLORATION: Writing a System of Linear Equations Work with a partner. Your family opens a bed-and-breakfast. They spend \( \$ 600 \) preparing a bedroom to rent. The cost to your family for food and utilities is \( \$ 15 \) per night. They charge \( \$ 75 \) per night to rent the bedroom. a. Write an equation that represents the costs. \[ \begin{array}{l} \text { Cost, } C \\ (\text { in dollars }) \end{array}=\begin{array}{l} \$ 15 \text { per } \\ \text { night } \end{array} \quad \begin{array}{c} \text { Number of } \\ \text { nights, } x \end{array}+\$ 600 \] b. Write an equation that represents the revenue (income). \[ \frac{\text { Revenue, } R}{(\text { in dollars })}=\begin{array}{l} \text { \$75 per } \\ \text { night } \end{array} \bullet \begin{array}{l} \text { Number of } \\ \text { nights, } x \end{array} \] c. A set of two (or more) linear equations is called a system of linear equations. Write the system of linear equations for this problem.

Let's dive into the world of equations! The system representing your family's bed-and-breakfast situation can be summarized as follows: 1. \( C = 15x + 600 \) (this captures the total costs incurred) 2. \( R = 75x \) (this details how much money is flowing in from rentals) So, your system of linear equations is: \[ \begin{cases} C = 15x + 600 \\ R = 75x \end{cases} \] Now, let's think about that bed-and-breakfast in a real-world context! If your family wants to break even, you'll need to figure out when \( C = R \). By substituting the equations, you can find the number of nights \( x \) needed to cover costs. This shows the practical application of solving systems of equations—making that B&B profitable and ensuring you enjoy pancakes while still making a profit! And remember, when writing your equations, keep an eye out for common mistakes. One common error is confusing costs with revenue; they have different meanings! Make sure you're distinguishing between them and aligning those equations properly. Happy mathing!