Final answer:
To solve the linear program, graph the boundary lines of the constraints, find the intersection points, and evaluate the objective function at these points to determine the minimum value.
Step-by-step explanation:
To solve the linear program algebraically, we need to minimize the objective function 3x_1 + 4x_2 subject to the constraints x_1 + 3x_2 ≥ 9 and 2x_1 + x_2 > 6, with both x_1, x_2 ≥ 0. First, we convert the inequalities to equations to find the boundary lines of the feasible region: x_1 + 3x_2 = 9 and 2x_1 + x_2 = 6. By graphing these lines and determining the feasible region they enclose, we can then identify the corner points where the boundaries intersect. The solution to the linear program will be at one of these corner points, so we evaluate the objective function at each corner point to find the minimum value.
The first step is to solve the equations for y in terms of x (where y is x_2 and x is x_1):
- For x_1 + 3x_2 = 9, x_2 equals (9 - x_1)/3.
- For 2x_1 + x_2 = 6, x_2 equals 6 - 2x_1.
Next, we plot these equations on a graph to find the intersection points, which are potential solutions. To ensure all points satisfy the original constraints, we check the inequality conditions. Finally, we plug in the values of x_1 and x_2 at the intersection points into the objective function to determine which gives the minimum value.