Final answer:
To find the optimal solution of a linear programming problem graphically, graph the constraints to find the feasible region, then plot the objective function to see where it is optimal within that region. Convert to standard form by modifying inequalities and variable constraints. Analytical methods are more accurate and can handle higher dimensions.
Step-by-step explanation:
To solve a linear programming problem using the graphical method, first, plot the constraint inequalities to find the feasible region. Then, graph the objective function and determine where it reaches its maximum or minimum value within the feasible region. This point is the optimal solution. The value of the objective function at this point is the optimal objective function value.
Converting a linear program to standard form consists of ensuring that all the inequalities are in the form of ≥ and all the variables are non-negative. If necessary, slack variables are added to convert inequalities into equalities.
Analytical techniques like the simplex method are more accurate than graphical methods because they are based on calculations rather than visual estimates. In graphical solutions, the accuracy is limited by how well you can draw and read the graph, and it is typically restricted to two-variable problems. Analytical methods do not have these limitations and can handle higher-dimensional problems.