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The question provided is about finding the solution set for the system of inequalities:

\[
\begin{aligned}
6x + 4y & \leq 24, \\
5x + 5y & \leq 25, \\
x & \geq 0, \\
y & \geq 0.
\end{aligned}
\]

The graph provided shows different regions labeled (i), (ii), (iii), (iv), and (v).

Let's analyze the system of inequalities:

1. **Rewrite the inequalities in slope-intercept form**:
    - \( 6x + 4y \leq 24 \) becomes \( y \leq -\frac{3}{2}x + 6 \),
    - \( 5x + 5y \leq 25 \) becomes \( y \leq -x + 5 \).

2. **Boundary lines**:
    - The line \( y = -\frac{3}{2}x + 6 \) has a slope of \(-\frac{3}{2}\) and intersects the \(y\)-axis at 6.
    - The line \( y = -x + 5 \) has a slope of \(-1\) and intersects the \(y\)-axis at 5.

3. **Non-negativity conditions**:
    - \( x \geq 0 \) and \( y \geq 0 \) restrict the solution to the first quadrant.

4. **Feasible region**:
    The feasible region is the area where all inequalities hold true, which corresponds to the area where:
    - The region is below both lines,
    - It lies in the first quadrant.

From the graph, it appears that region **(iii)** satisfies all these conditions. It lies below both boundary lines and is in the first quadrant.

Thus, the correct answer is:

\[
\boxed{(iii)}
\]

Would you like further details or clarifications? Here are some follow-up questions for deeper understanding:

1. How do we determine the slope and intercept from an inequality?
2. What happens to the region when the inequalities involve strict inequalities (like \( < \) instead of \( \leq \))?
3. How does the first quadrant restriction affect the solution set?
4. How can graphing software or a calculator assist in visualizing inequalities?
5. What is the significance of boundary lines in linear programming?

**Tip**: When solving systems of inequalities, always remember to consider the constraints \( x \geq 0 \) and \( y \geq 0 \) if given, as they significantly narrow down the feasible region.

Himpunan Penyelesaian dari: 6x + 4y ≤ 24; 5x + 5y ≤ 25; x ≥ 0; y ≥ 0

Learn how to find the solution set for the system of inequalities 6x + 4y ≤ 24, 5x + 5y ≤ 25, x ≥ 0, y ≥ 0 using a graphical method. This example demonstrates how to identify the feasible region in a graph of inequalities.

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