ART TEACHES MATHEMATICS IN THE MODERN WORLD: LESSON 3-GRAPHICAL METHOD (PART II)

In this class on mathematics in the modern world, we discuss solving a linear programming maximization problem using a graphical method. The problem has two decision variables and four explicit constraints, plus implicit non-negativity constraints ensuring the decision variables are non-negative. The objective is to maximize P = 7x + 5y under these constraints.

The first explicit constraint is represented as 5x + 8y ≤ 14. To solve it, we initially treat the inequality as an equation to find intercepts, where the line will intersect the axes. By setting y=0, we find the x-intercept as x=8, and by setting x=0, we derive the y-intercept as y=5. This helps in plotting the line representing the boundary of the constraint.

A parallel approach is employed on the second constraint represented by 9x + 6y ≤ 54. Similar to the first constraint, we first find the x-intercept (x=6) and y-intercept (y=9) by setting y=0 and x=0 respectively. The line is drawn as the boundary, below which lies the feasible region for this constraint.

The third constraint where x ≥ 2 is tackled, depicting a vertical line on the Cartesian plane at x=2. The fourth constraint, y ≤ 4, is represented by a horizontal line at y=4. The shaded areas below and to the right of these lines respectively indicate the feasible solution spaces aligning with each constraint.

Once all constraints are plotted, the intersecting lines form a bounded feasible region on the Cartesian plane. The objective function’s value reaches its maximum somewhere in this region, and the graphical method entails evaluating this function at each intersection point or vertex of the polygon shaped by the constraints.

A structured process is followed to evaluate each vertex of the feasible region. Each point's coordinates (x, y) are substituted into the objective function P = 7x + 5y, allowing us to calculate the corresponding P values. This procedure identifies potential maximum values of the objective function.

As each vertex is evaluated, initial calculations suggest multiple feasible solutions, though none optimal yet. The best solution maximizes P by identifying the intersection points of the lines, solving the simultaneous linear equations representing the intersecting constraints.

The vertex where the lines intersect provides the coordinates that furnish the highest P value so far. Further analysis requires solving equations of form 9x + 6y = 54 and 5x + 8y = 14 to precisely locate intersection points and thus compute P at these vertices.

By systematically calculating values of P for each intersection point of the constraints, the vertex (4.57, 2.14) yields the highest P value of 42.69, thus identified as the optimal solution. This point represents the best combination of decision variable values under the given conditions.

Recapping, various solution points for decision variables x and y met the constraints, but only the vertex (4.57, 2.14) resulted in the highest value of P, concluding the exercise of graph-based maximization for the problem.

The final feasible solution yields x = 4.57, y = 2.14 with a maximum P of 42.69, showcasing this approach's efficacy in linear programming. The method exemplifies how graphical solution processes clarify optimal solutions amidst multiple constraint conditions.

In the next lesson, the focus will shift to minimization problems by graphical methods. Audience questions are encouraged through comments, ensuring interactive learning, while subscribing to the channel is recommended for continuous engagement with mathematical techniques.