minimize 4 x+5 y for the bounded feasible region displayed above

To minimize the objective function \(4x + 5y\) over a bounded feasible region, we'll need to follow these steps:

### Step 1: Identify the system of inequalities that define the feasible region
This region is usually defined by a set of linear inequalities. If this system was provided earlier (or in your description), it would have constraints like:

\[
11x + 8y \leq 500 \quad \text{(assembly constraint)}
\]
\[
2x + y \leq 110 \quad \text{(finishing constraint)}
\]
\[
x \geq 0 \quad \text{and} \quad y \geq 0 \quad \text{(non-negativity constraints)}
\]

If these inequalities define the feasible region, we need to identify the **corner points** (or vertices) of this region.

### Step 2: Find the corner points
The corner points (vertices) of the feasible region are the points where the boundary lines of the inequalities intersect. To find them, we solve pairs of equations formed by the boundaries of the inequalities.

1. **Solve** \(11x + 8y = 500\) and \(2x + y = 110\) (assembly and finishing constraints).
2. **Find intersections** with axes (e.g., \(x = 0\) or \(y = 0\)) to find other corner points.

Let's start by solving the two constraint equations together.

#### a) Solve \(11x + 8y = 500\) and \(2x + y = 110\)

From \(2x + y = 110\), solve for \(y\):

\[
y = 110 - 2x
\]

Substitute into the first equation \(11x + 8y = 500\):

\[
11x + 8(110 - 2x) = 500
\]
\[
11x + 880 - 16x = 500
\]
\[
-5x + 880 = 500
\]
\[
-5x = -380
\]
\[
x = 76
\]

Now, substitute \(x = 76\) back into \(y = 110 - 2x\):

\[
y = 110 - 2(76) = 110 - 152 = -42
\]

This gives the intersection point \((x, y) = (76, -42)\). However, because \(y = -42\) is negative and the feasible region requires \(y \geq 0\), this point is outside the feasible region.

#### b) Find intersections with the axes

We need to find valid corner points where the lines intersect the positive axes (\(x \geq 0\), \(y \geq 0\)):

- When \(x = 0\) in \(2x + y = 110\), we get \(y = 110\).
- When \(y = 0\) in \(2x + y = 110\), we get \(2x = 110\

Minimize 4x + 5y for the bounded feasible region displayed above

Learn how to minimize the objective function 4x + 5y for a bounded feasible region defined by inequality constraints. This problem covers linear programming concepts, the corner-point method, and optimization strategies suitable for students in Grades 11-12.

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