To solve the given linear programming problem (LPP) graphically, we will follow these steps:

Step 1: Problem Formulation

Objective: Maximize $Z = 3x_1 + 7x_2$

Subject to the following constraints:

1. $3x_1 + 5x_2 \leq 20$
2. $x_1 + 3x_2 \leq 9$
3. $x_1 \geq 2$
4. $x_1 \geq 0, \, x_2 \geq 0$

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Step 2: Convert Inequalities to Equalities

1. $3x_1 + 5x_2 = 20$ (Constraint 1)
2. $x_1 + 3x_2 = 9$ (Constraint 2)
3. $x_1 = 2$ (Constraint 3)

Step 3: Plot Each Constraint

For graphical representation, we will find the intercepts for each constraint:

1. $3x_1 + 5x_2 = 20$

   • When $x_1 = 0$, $5x_2 = 20 \Rightarrow x_2 = 4$.
   • When $x_2 = 0$, $3x_1 = 20 \Rightarrow x_1 = \frac{20}{3} \approx 6.67$.

2. $x_1 + 3x_2 = 9$

   • When $x_1 = 0$, $3x_2 = 9 \Rightarrow x_2 = 3$.
   • When $x_2 = 0$, $x_1 = 9$.

3. $x_1 = 2$ is a vertical line passing through $x_1 = 2$.

4. Non-negativity constraints ensure that the solution lies in the first quadrant.

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Step 4: Identify Feasible Region

The feasible region is the intersection of all constraints and lies in the first quadrant. We need to find the corner points of this region by solving the following system of equations:

1. Intersection of $3x_1 + 5x_2 = 20$ and $x_1 + 3x_2 = 9$:

   • Solve the system simultaneously using substitution or elimination.

   Multiply the second equation by 3: $3x_1 + 9x_2 = 27$

   Subtract the first equation from this result: $(3x_1 + 9x_2) - (3x_1 + 5x_2) = 27 - 20 \Rightarrow 4x_2 = 7 \Rightarrow x_2 = \frac{7}{4} = 1.75$

   Substitute $x_2 = 1.75$ into $x_1 + 3x_2 = 9$: $x_1 + 3(1.75) = 9 \Rightarrow x_1 = 9 - 5.25 = 3.75$

   Intersection point: $(3.75, 1.75)$.

2. Intersection of $x_1 = 2$ and $3x_1 + 5x_2 = 20$:

   • Substitute $x_1 = 2$ into $3x_1 + 5x_2 = 20$: $3(2) + 5x_2 = 20 \Rightarrow 5x_2 = 14 \Rightarrow x_2 = \frac{14}{5} = 2.8$

   Intersection point: $(2, 2.8)$.

3. Intersection of $x_1 = 2$ and $x_1 + 3x_2 = 9$:

   • Substitute $x_1 = 2$ into $x_1 + 3x_2 = 9$: $2 + 3x_2 = 9 \Rightarrow 3x_2 = 7 \Rightarrow x_2 = \frac{7}{3} \approx 2.33$

   Intersection point: $(2, 2.33)$.

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Step 5: Evaluate the Objective Function

We now evaluate $Z = 3x_1 + 7x_2$ at each corner point:

1. At $(3.75, 1.75)$: $Z = 3(3.75) + 7(1.75) = 11.25 + 12.25 = 23.5$

2. At $(2, 2.8)$: $Z = 3(2) + 7(2.8) = 6 + 19.6 = 25.6$

3. At $(2, 2.33)$: $Z = 3(2) + 7(2.33) = 6 + 16.33 = 22.33$

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Step 6: Conclusion

The maximum value of $Z$ occurs at $(2, 2.8)$ with $Z = 25.6$.

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Do you want me to plot this graph or explain anything in more detail?

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Related Questions:

1. How do we solve LPPs using the Simplex method?
2. What are the benefits of solving LPPs graphically vs. algebraically?
3. How can we interpret corner points in the context of optimization?
4. How do non-negativity constraints impact the feasible region?
5. Can we solve LPPs with more than two variables graphically?

Tip: Always double-check your intersection points when solving LPPs graphically to ensure accuracy.