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

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

\[
\begin{align*}
x + y &= 10 \quad \text{(1)} \\
x &\geq 2 \quad \text{(2)} \\
y &\geq 3 \quad \text{(3)} \\
x &\geq 0 \quad \text{(4)} \\
y &\geq 0 \quad \text{(5)}
\end{align*}
\]

### Steps to Solve:

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

2. **Apply the other constraints to \( y \):**
   Using \( y \geq 3 \):
   \[
   10 - x \geq 3
   \]
   Solving this inequality:
   \[
   x \leq 7
   \]

   Now we have the constraint \( 2 \leq x \leq 7 \).

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

4. **Minimize the objective function:**
   Since \( Z = -2x + 70 \), the value of \( Z \) decreases as \( x \) increases. To minimize \( Z \), we need to maximize \( x \).

   From the constraint \( 2 \leq x \leq 7 \), the maximum value of \( x \) is 7.

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

Thus, the minimum value of \( Z \) is **56**, which occurs at \( x = 7 \) and \( y = 3 \).

Would you like further clarification or details?

Here are 5 related questions to explore:

1. What happens to the value of \( Z \) if the coefficients of \( x \) or \( y \) change?
2. How would the solution differ if we were maximizing \( Z \) instead of minimizing?
3. What graphical methods can be used to solve linear programming problems?
4. How do changes in the constraints affect the optimal solution?
5. How would the solution change if the constraint \( x + y = 10 \) were an inequality instead?

**Tip:** In linear programming, always check the boundaries of the feasible region, as optimal solutions often lie at these boundaries.

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

Solve the linear programming problem: Minimize Z = 5x + 7y with constraints x + y = 10, x >= 2, y >= 3. Follow step-by-step to learn how to find the optimal solution for this system of linear inequalities.

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