Question

The optimal value for the linear programming problem Maximize: \( 6x_1 + 5x_2 \) subject to: \[ 3x_1 + 2x_2 \leq 12 \] \[ -x_1 + x_2 \leq 1 \] \[ x_1, x_2 \geq 0 \] is __________.

Hint:

To solve a linear programming problem using the graphical method, plot the constraints and identify the feasible region. Evaluate the objective function at the corner points to find the optimal value.

Answer 1

Correct Answer: 27

We are asked to find the optimal value for the given linear programming problem. To solve this, we will use the graphical method. Step 1: Graph the constraints. The constraints are: \[ 3x_1 + 2x_2 \leq 12 \] \[ -x_1 + x_2 \leq 1 \] \[ x_1, x_2 \geq 0 \] - For \( 3x_1 + 2x_2 = 12 \), when \( x_1 = 0 \), \( x_2 = 6 \), and when \( x_2 = 0 \), \( x_1 = 4 \). - For \( -x_1 + x_2 = 1 \), when \( x_1 = 0 \), \( x_2 = 1 \), and when \( x_2 = 0 \), \( x_1 = -1 \), but since \( x_1 \geq 0 \), the feasible region does not include negative values for \( x_1 \). Step 2: Identify the feasible region. The feasible region is the area that satisfies all the inequalities, and it is bounded by the lines \( 3x_1 + 2x_2 = 12 \) and \( -x_1 + x_2 = 1 \), and the axes \( x_1 \geq 0 \) and \( x_2 \geq 0 \). Step 3: Find the corner points. The corner points (vertices) of the feasible region are the points where the constraint lines intersect: - The intersection of \( 3x_1 + 2x_2 = 12 \) and \( -x_1 + x_2 = 1 \): Solving these equations simultaneously: \[ 3x_1 + 2x_2 = 12 \] \[ -x_1 + x_2 = 1 \quad \Rightarrow \quad x_2 = x_1 + 1 \] Substitute \( x_2 = x_1 + 1 \) into \( 3x_1 + 2x_2 = 12 \): \[ 3x_1 + 2(x_1 + 1) = 12 \] \[ 3x_1 + 2x_1 + 2 = 12 \] \[ 5x_1 = 10 \quad \Rightarrow \quad x_1 = 2 \] Substituting \( x_1 = 2 \) into \( x_2 = x_1 + 1 \): \[ x_2 = 2 + 1 = 3 \] So, the intersection point is \( (2, 3) \). - The other two corner points are \( (0, 0) \) and \( (4, 0) \) (from the constraints). Step 4: Evaluate the objective function at the corner points. We evaluate the objective function \( 6x_1 + 5x_2 \) at the corner points: - At \( (0, 0) \), \( 6x_1 + 5x_2 = 6(0) + 5(0) = 0 \) - At \( (4, 0) \), \( 6x_1 + 5x_2 = 6(4) + 5(0) = 24 \) - At \( (2, 3) \), \( 6x_1 + 5x_2 = 6(2) + 5(3) = 12 + 15 = 27 \) Step 5: Conclusion. The maximum value of the objective function is 27, which occurs at the point \( (2, 3) \). Final Answer: Thus, the optimal value is \( \boxed{27} \).