Min Z = 5x + 7y

S. T

x + y = 10

x ≥ 2

y ≥ 3

x ≥ 0

x ≥ 0

We are tasked with minimizing the objective function \( Z = 5x + 7y \), subject to the following constraints:

\[
\begin{aligned}
x + y &= 10 \\
x &\geq 2 \\
y &\geq 3 \\
x &\geq 0 \\
y &\geq 0 \\
\end{aligned}
\]

### Steps to solve:

1. **Substitute from the equality constraint**:
   From \( x + y = 10 \), we can express \( y \) in terms of \( x \) as:
   \[
   y = 10 - x
   \]

2. **Update the constraints**:
   Since \( y = 10 - x \), we need to ensure that this satisfies the other constraints:
   \[
   x \geq 2 \quad \text{and} \quad y = 10 - x \geq 3
   \]
   
   Solving \( 10 - x \geq 3 \):
   \[
   10 - 3 \geq x \implies x \leq 7
   \]

   Therefore, the feasible range for \( x \) is:
   \[
   2 \leq x \leq 7
   \]

3. **Express the objective function**:
   Substitute \( y = 10 - x \) into the objective function \( Z = 5x + 7y \):
   \[
   Z = 5x + 7(10 - x) = 5x + 70 - 7x = -2x + 70
   \]

4. **Minimize \( Z \)**:
   Since the objective function is linear and decreasing in \( x \), we minimize \( Z \) by choosing the maximum value of \( x \), which is \( x = 7 \).

5. **Calculate \( Z \) at \( x = 7 \)**:
   If \( x = 7 \), then:
   \[
   y = 10 - 7 = 3
   \]
   Substituting into the objective function:
   \[
   Z = 5(7) + 7(3) = 35 + 21 = 56
   \]

### Conclusion:
The minimum value of \( Z \) is **56**, which occurs at \( x = 7 \) and \( y = 3 \).

---

Would you like more details or have any questions?

Here are 5 related questions for further understanding:

1. What happens to \( Z \) if \( x = 2 \) instead of 7?
2. How do we solve a maximization problem using the same approach?
3. Can you solve a minimization problem with more constraints?
4. How does the geometry of the feasible region affect the solution?
5. What if the constraint \( x + y = 10 \) were an inequality?

**Tip:** In optimization problems, always check if the objective function is increasing or decreasing to identify boundary solutions effectively.

Min Z = 5x + 7y
S.T
x + y = 10
x ≥ 2
y ≥ 3
x ≥ 0
y ≥ 0

Explore how to minimize the objective function Z = 5x + 7y with linear constraints x + y = 10, x ≥ 2, and y ≥ 3. This problem introduces linear programming and optimization techniques, making it ideal for students in Grades 10-12.

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