To solve the linear programming problem, we need to use the standard methods for optimization while taking into account the provided constraints. The goal is to minimize the objective function [tex]\( z = 4x_1 + x_2 \)[/tex].
Let's start by rewriting the problem:
### Objective
[tex]\[ \text{Minimize } z = 4x_1 + x_2 \][/tex]
### Subject to
[tex]\[
\begin{aligned}
3x_1 + x_2 &= 30 \quad \text{(Constraint 1)} \\
4x_1 + 3x_2 &> 60 \quad \text{(Constraint 2)} \\
x_1 + 2x_2 &\leq 40 \quad \text{(Constraint 3)} \\
x_1, x_2 &\geq 0 \quad \text{(Non-negativity)}
\end{aligned}
\][/tex]
Before we proceed, let's address the inequalities to convert them into standard form:
- The second constraint, [tex]\(4x_1 + 3x_2 > 60\)[/tex], can be rewritten as [tex]\(-4x_1 - 3x_2 \leq -60\)[/tex].
- The third constraint, [tex]\(x_1 + 2x_2 \leq 40\)[/tex], remains as is.
Now, let's summarize these constraints in the standard form compatible with optimization techniques:
### Constraints in Standard Form:
[tex]\[
\begin{aligned}
3x_1 + x_2 &= 30 \quad \text{(Equality Constraint)} \\
-4x_1 - 3x_2 &\leq -60 \quad \text{(Inequality Constraint 1)} \\
x_1 + 2x_2 &\leq 40 \quad \text{(Inequality Constraint 2)} \\
x_1, x_2 &\geq 0 \quad \text{(Bounds)}
\end{aligned}
\][/tex]
Given these constraints and the objective function, we can solve the optimization problem.
### Solution:
After solving the problem step-by-step using the appropriate linear programming methods, we find that the optimal solution for the decision variables is:
[tex]\[ x_1 = 4 \][/tex]
[tex]\[ x_2 = 18 \][/tex]
Substituting these values back into the objective function yields:
[tex]\[ z = 4(4) + 18 = 16 + 18 = 34 \][/tex]
Thus, the minimized value of the objective function [tex]\( z \)[/tex] is:
[tex]\[ z = 34 \][/tex]
### Conclusion:
The optimal solution to the given linear programming problem is:
[tex]\[ x_1 = 4 \][/tex]
[tex]\[ x_2 = 18 \][/tex]
And the minimized value of [tex]\( z \)[/tex] is:
[tex]\[ z = 34 \][/tex]
