https://file.aiquickdraw.com/mathbot/file/1728841475814fjxpyjoe.jpg

The image shows a system of linear inequalities that needs to be graphed on a coordinate plane. The inequalities are as follows:

\[
\begin{cases} 
12x + 10y \leq 120 \\
9x + 13y \leq 117 \\
x \geq 0 \\
y \geq 0 
\end{cases}
\]

### Let's break down the steps to solve and graph the system:

1. **Rewrite the inequalities as equalities to find boundary lines:**
   - First inequality: \(12x + 10y = 120\)
   - Second inequality: \(9x + 13y = 117\)

2. **Find the intercepts for each line** to graph them.

   - **For \(12x + 10y = 120\):**
     - \(x\)-intercept: Set \(y = 0\), then \(12x = 120 \implies x = 10\).
     - \(y\)-intercept: Set \(x = 0\), then \(10y = 120 \implies y = 12\).

   - **For \(9x + 13y = 117\):**
     - \(x\)-intercept: Set \(y = 0\), then \(9x = 117 \implies x = 13\).
     - \(y\)-intercept: Set \(x = 0\), then \(13y = 117 \implies y = 9\).

3. **Graph each boundary line:**
   - Plot the points \((10, 0)\) and \((0, 12)\) for the first line, and connect them.
   - Plot the points \((13, 0)\) and \((0, 9)\) for the second line, and connect them.

4. **Determine the solution region:**
   - For \(12x + 10y \leq 120\), shade the region below or on the line.
   - For \(9x + 13y \leq 117\), shade the region below or on the line.
   - The constraints \(x \geq 0\) and \(y \geq 0\) indicate that the region of interest lies in the first quadrant.

5. **Find the feasible region**, which is the intersection of the shaded regions from both inequalities in the first quadrant.

Would you like me to provide a detailed graph of this solution? Let me know if you have any questions or need further clarifications!

### Follow-up Questions:
1. What is the significance of the intercepts in graphing linear inequalities?
2. How do you identify the solution region of a system of inequalities?
3. How would you approach solving the system if it had three inequalities instead of two?
4. Why is it important that \(x \geq 0\) and \(y \geq 0\) in this problem?
5. What happens if the inequalities were strict (e.g., \(<\) instead of \(\leq\))?

### Tip:
When solving linear inequalities, always test a point (like the origin) to check which side of the boundary line should be shaded.

Graph the system of inequalities:
1. 12x + 10y ≤ 120
2. 9x + 13y ≤ 117
3. x ≥ 0
4. y ≥ 0

Learn how to graph a system of linear inequalities, including finding intercepts and determining the feasible region. This problem involves two inequalities, x and y non-negative constraints, and covers concepts of graphing and shading. Suitable for students in Grades 9-11.

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