Questions: x+y ≤ 40 x ≥ 5 y ≥ 10 100x+200y ≤ 7,000

Solution Steps

To solve this system of inequalities, we need to find the feasible region that satisfies all the given constraints. This involves checking each inequality and determining the set of points (x, y) that satisfy all conditions simultaneously. We can use Python to plot these inequalities and visually identify the feasible region.

To solve the given system of inequalities, we will follow a step-by-step approach to find the feasible region and determine the solution.

The system of inequalities is:

1. \( x + y \leq 40 \)
2. \( x \geq 5 \)
3. \( y \geq 10 \)
4. \( 100x + 200y \leq 7,000 \)

We will graph each inequality on the coordinate plane to find the feasible region.

1. Graph \( x + y \leq 40 \):

   • The line \( x + y = 40 \) is a boundary.
   • To find intercepts, set \( x = 0 \) to get \( y = 40 \) and set \( y = 0 \) to get \( x = 40 \).
   • The region below this line is the solution for \( x + y \leq 40 \).

2. Graph \( x \geq 5 \):

   • This is a vertical line at \( x = 5 \).
   • The region to the right of this line is the solution for \( x \geq 5 \).

3. Graph \( y \geq 10 \):

   • This is a horizontal line at \( y = 10 \).
   • The region above this line is the solution for \( y \geq 10 \).

4. Graph \( 100x + 200y \leq 7,000 \):

   • Simplify to \( x + 2y \leq 70 \).
   • The line \( x + 2y = 70 \) is a boundary.
   • To find intercepts, set \( x = 0 \) to get \( y = 35 \) and set \( y = 0 \) to get \( x = 70 \).
   • The region below this line is the solution for \( x + 2y \leq 70 \).

The feasible region is the intersection of all the regions defined by the inequalities. It is bounded by the lines:

• \( x + y = 40 \)
• \( x = 5 \)
• \( y = 10 \)
• \( x + 2y = 70 \)

To find the vertices, solve the system of equations formed by the intersection of the lines:

1. Intersection of \( x + y = 40 \) and \( x = 5 \): \[ 5 + y = 40 \implies y = 35 \] Vertex: \( (5, 35) \)

2. Intersection of \( x + y = 40 \) and \( y = 10 \): \[ x + 10 = 40 \implies x = 30 \] Vertex: \( (30, 10) \)

3. Intersection of \( x = 5 \) and \( x + 2y = 70 \): \[ 5 + 2y = 70 \implies 2y = 65 \implies y = 32.5 \] Vertex: \( (5, 32.5) \)

4. Intersection of \( y = 10 \) and \( x + 2y = 70 \): \[ x + 2(10) = 70 \implies x + 20 = 70 \implies x = 50 \] Vertex: \( (50, 10) \)

Final Answer