1. Solve the following L.P.P. graphically
Min Z = 6x + 2y
Such that x + 2y 30
x + 2y 120
2x + y 40
x 0, y .

It seems like the inequality symbols in the constraints are missing. From the context, I will assume the constraints should look something like this:

\[
\text{Minimize } Z = 6x + 2y
\]
\[
\text{Subject to:}
\]
1. \( x + 2y \geq 30 \)
2. \( x + 2y \leq 120 \)
3. \( 2x + y \leq 40 \)
4. \( x \geq 0 \), \( y \geq 0 \)

Let's proceed to solve this Linear Programming Problem (LPP) graphically.

### Steps to solve graphically:
1. **Plot the lines for the constraints**:
   - For \( x + 2y = 30 \) and \( x + 2y = 120 \), find the points of intersection with the axes.
   - For \( 2x + y = 40 \), find the points of intersection with the axes.

2. **Shade the feasible region**:
   - Identify the region that satisfies all the inequalities.

3. **Identify the corner points**:
   - The feasible region will have corner points where the lines intersect.

4. **Evaluate the objective function**:
   - Substitute the corner points into \( Z = 6x + 2y \) to find the minimum value.

I'll walk you through the solution and plot it graphically. Let's begin by solving for the points where the lines intersect the axes.

### Step 1: Find the intercepts

#### For \( x + 2y = 30 \):
- When \( x = 0 \), \( 2y = 30 \) → \( y = 15 \).
- When \( y = 0 \), \( x = 30 \).

Thus, the intercepts are \( (30, 0) \) and \( (0, 15) \).

#### For \( x + 2y = 120 \):
- When \( x = 0 \), \( 2y = 120 \) → \( y = 60 \).
- When \( y = 0 \), \( x = 120 \).

Thus, the intercepts are \( (120, 0) \) and \( (0, 60) \).

#### For \( 2x + y = 40 \):
- When \( x = 0 \), \( y = 40 \).
- When \( y = 0 \), \( 2x = 40 \) → \( x = 20 \).

Thus, the intercepts are \( (20, 0) \) and \( (0, 40) \).

### Step 2: Plot the lines and identify the feasible region
Now we will plot these lines on a graph and find the feasible region that satisfies all the constraints.

Let me plot this graphically.

Solve the following L.P.P. graphically: Min Z = 6x + 2y such that x + 2y ≥ 30, x + 2y ≤ 120, 2x + y ≤ 40, x ≥ 0, y ≥ 0.

Learn how to solve a linear programming problem graphically with the objective to minimize Z = 6x + 2y, subject to constraints. Follow a step-by-step guide to plotting the feasible region and identifying the corner points for optimization. Suitable for high school students (Grades 10-12).

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