https://file.aiquickdraw.com/mathbot/file/17286328233434p3zaio9.jpg

The question in the image asks to "determine the system of inequalities from the solution region shown."

### Step-by-Step Process:

1. **Identifying the boundary lines:**
   - There are two lines in the graph. Let's find their equations based on the given points.
   
   **First line** (left side):
   - Passes through the points (0, 10) and (10, 0).
   - The slope \( m = \frac{0 - 10}{10 - 0} = -1 \).
   - Using the point-slope form of the line equation \( y = mx + c \), with \( m = -1 \) and the intercept \( c = 10 \), the equation is:
     \[
     y = -x + 10
     \]

   **Second line** (right side):
   - Passes through the points (0, 1) and (5, 0).
   - The slope \( m = \frac{0 - 1}{5 - 0} = -\frac{1}{5} \).
   - Using the point-slope form with \( m = -\frac{1}{5} \) and intercept \( c = 1 \), the equation is:
     \[
     y = -\frac{1}{5}x + 1
     \]

2. **Inequalities based on the shaded region:**
   - The shaded region is below both of these lines.
   
   - For the first line \( y = -x + 10 \), the shaded region is **below** the line, so the inequality is:
     \[
     y \leq -x + 10
     \]
   
   - For the second line \( y = -\frac{1}{5}x + 1 \), the shaded region is also below the line, giving the inequality:
     \[
     y \leq -\frac{1}{5}x + 1
     \]

3. **Non-negative constraints:**
   - The region is only in the first quadrant, so the constraints on the variables are:
     \[
     x \geq 0, \quad y \geq 0
     \]

### Final System of Inequalities:
\[
\begin{aligned}
y &\leq -x + 10 \\
y &\leq -\frac{1}{5}x + 1 \\
x &\geq 0 \\
y &\geq 0
\end{aligned}
\]

This is the system of inequalities representing the shaded region in the graph.

Would you like more details on any part of the solution? Feel free to ask! Here are 5 related questions to explore further:

1. How can we verify if a point lies within the solution region of a system of inequalities?
2. What is the significance of the slope in the equations of lines?
3. How do the inequalities change if the region is above the lines instead of below?
4. Can we solve this system graphically without drawing the actual graph?
5. How do we determine the intersection points of two lines?

**Tip:** When dealing with inequalities, always check whether the shaded region is above or below the boundary line to set the correct inequality.

Tentukan sistem pertidaksamaan dari daerah himpunan penyelesaian diatas.

Learn how to determine a system of linear inequalities from a shaded region in a graph. This example covers graphing, linear inequalities, and constraints in the first quadrant. Suitable for students in Grades 9-11.

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