Final answer:
To solve the linear programming problem graphically, plot each inequality as a line, find the feasible region, calculate the objective function at each vertex within that region, and determine which coordinates maximize the objective function.
Step-by-step explanation:
Firstly, let's write down the constraints for the linear programming problem:
- x₁ + 2x₂ ≤ 6
- 2x₁ + 3x₂ ≤ 10
- x₁ ≤ 2
- x₂ ≥ 1
- x₁, x₂ ≥ 0
Next, we will graph these inequalities on a coordinate plane.
- For the equation x₁+2x₂=6, plot the intercepts (6,0) and (0,3) and draw the line, shading towards the origin to represent the ≤ inequality.
- For 2x₁+3x₂=10, plot the intercepts (5,0) and (0,⅔3) and draw the line, again shading towards the origin.
- For x₁=2, draw a vertical line at x₁=2 and shade to the left.
- For x₂≥1, draw a horizontal line at x₂=1 and shade upward.
Now we identify the feasible region, which is where all the shadings overlap. Then, we calculate the value of the objective function Z=3x₁+2x₂ at each vertex of the feasible region. The coordinates that give the maximum Z within the feasible region will be the solution to the problem.
Finally, label the axes with x₁ and x₂ and scale them so they accommodate the feasible region, which is determined by the points of intersection of the lines. In this problem, you can set the scale up to at least (2,3) to capture all possible solutions.