Graphing Constraint Lines and Solutions in Linear Programming

In the realm of high school mathematics, understanding the concept of linear programming is crucial. One of the key aspects of linear programming is the ability to visualize the constraints and the solutions that satisfy them. In this article, we will explore the process of graphing constraint lines and the solutions that satisfy each constraint.

Constraint Line a: 3A + 2B ≤ 18

To graph the constraint line for the inequality 3A + 2B ≤ 18, we need to first rearrange the equation to solve for one of the variables in terms of the other. In this case, we can solve for B in terms of A:

3A + 2B ≤ 18
2B ≤ 18 - 3A
B ≤ 9 - (3/2)A

Now, we can plot the constraint line on a coordinate plane. The slope of the line is -3/2, and the y-intercept is 9. The shaded region on the graph represents the solutions that satisfy this constraint.

Constraint Line b: 12A + 8B ≤ 480

Similarly, for the inequality 12A + 8B ≤ 480, we can rearrange the equation to solve for B in terms of A:

12A + 8B ≤ 480
8B ≤ 480 - 12A
B ≤ 60 - (3/2)A

The slope of this constraint line is -3/2, and the y-intercept is 60. The shaded region on the graph represents the solutions that satisfy this constraint.

Constraint Line c: 5A + 10B = 200

In this case, the constraint is an equality, so we can simply plot the line on the graph. The equation can be rearranged as:

5A + 10B = 200
B = 20 - (1/2)A

The slope of this constraint line is -1/2, and the y-intercept is 20. The solutions that satisfy this constraint lie on the line.

Combining the Constraints

Now, we can combine all three constraint lines on a single graph. The shaded region where all three constraints intersect represents the feasible region, which contains the solutions that satisfy all the constraints.

By graphing the constraint lines and the solutions that satisfy each constraint, you can gain a deeper understanding of the linear programming problem and the optimal solution that can be found within the feasible region.