Which of the following shows the correct graph of the feasible region for the following constraints?

5x + 5y ≤ 390

-x + y ≤ 10

x + 3y ≤ 90

x ≥ 0

y ≥ 0



Question 6 options:

Graph A


Graph B


Graph C


Graph D

To determine which graph shows the correct feasible region for the given system of inequalities, we need to analyze and plot each constraint step by step. Let's go through each constraint and its corresponding line:

1. **5x + 5y ≤ 390**
   - Simplify this to: \( x + y \leq 78 \)
   - The line \( x + y = 78 \) is the boundary, and the region below or on this line is the feasible region (since it's a "less than or equal to" inequality).

2. **-x + y ≤ 10**
   - Rearranging this gives: \( y \leq x + 10 \)
   - The line \( y = x + 10 \) is the boundary, and the region below this line is the feasible region.

3. **x + 3y ≤ 90**
   - The line \( x + 3y = 90 \) is the boundary, and the region below or on this line is the feasible region.

4. **x ≥ 0**
   - This is the constraint that ensures the region lies on or to the right of the y-axis.

5. **y ≥ 0**
   - This constraint ensures that the region lies on or above the x-axis.

### Steps for Graphing:
- First, plot each line representing the boundary of the inequalities.
- Shade the region that satisfies each inequality.
- The feasible region is where all these shaded areas overlap.

Given these constraints, the feasible region will be in the first quadrant (since \( x \geq 0 \) and \( y \geq 0 \)) and will be bounded by the lines formed from the equations above.

To determine which graph (A, B, C, or D) corresponds to this feasible region, you'd need to look for the intersection of all the shaded regions that meet these constraints. Without seeing the graphs, it's essential to match the graph that satisfies all these inequalities, especially focusing on how the lines interact and where the shaded region lies.

### Do you want me to explain further how to graph this step by step, or are you able to share an image of the graphs? 

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**5 Related Questions:**

1. What is the significance of the feasible region in linear programming?
2. How does the boundary line equation help in plotting constraints on a graph?
3. Can you have an unbounded feasible region, and what does that mean?
4. How do corner points of the feasible region help in optimizing linear objective functions?
5. What are the implications if the feasible region has no solution?

**Tip:** Always double-check your inequality signs when plotting, as they determine whether to shade above or below the boundary line.

Which of the following shows the correct graph of the feasible region for the following constraints? 5x + 5y ≤ 390, -x + y ≤ 10, x + 3y ≤ 90, x ≥ 0, y ≥ 0. Question 6 options: Graph A, Graph B, Graph C, Graph D

Learn how to graph the feasible region for the system of inequalities: 5x + 5y ≤ 390, -x + y ≤ 10, x + 3y ≤ 90, x ≥ 0, y ≥ 0. Understand how to apply the boundary line method and determine the overlap of shaded regions for the correct graph. Suitable for Grades 9-12.

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