Linear programming: A Geometric Approach
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Linear programming problems involve maximizing or minimizing a linear objective function subject to linear inequality constraints. These problems can be analyzed geometrically for problems with two variables. The solution set of a linear equation in two variables forms a straight line, while a linear inequality forms either the half-plane on one side of the line or the other, depending on whether a test point satisfies the inequality. To graph a linear inequality, one first graphs the corresponding equality and then tests a point to determine which half-plane is the solution set.
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This document discusses various direct methods for solving linear systems of equations, including graphical methods, Cramer's rule, elimination of unknowns, Gaussian elimination, Gaussian-Jordan elimination, and LU decomposition. It provides examples and explanations of each method. Graphical methods can solve systems of 2 equations visually by plotting the lines. Cramer's rule uses determinants to find solutions. Elimination of unknowns combines equations to remove variables. Gaussian elimination converts the matrix to upper triangular form. Gaussian-Jordan elimination converts it to an identity matrix. LU decomposition factors the matrix into lower and upper triangular matrices.
Direct methods
1) The graphical method involves graphing the lines represented by each equation on the same coordinate plane and finding the point where they intersect, which gives the solution.
2) Cramer's rule expresses each unknown as a ratio of determinants, with the numerator being the determinant of the coefficient matrix with one column replaced by the constants.
3) Gaussian elimination transforms the coefficient matrix into upper triangular form using elementary row operations, then back substitution solves for the unknowns.
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Linear programming: A Geometric Approach
- 1. Linear Programming: A Geometric Approach EGA
- 2. • MANY PRACTICAL PROBLEMS involve maximizing or minimizing a function subject to certain constraints. • For example, we might wish to maximize a profit function subject to certain limitations on the amount of material and labor available. • Maximization or minimization problems that can be formulated in terms of a linear objective function and constraints in the form of linear inequalities are called linear programming problems. • In this discussion, we look at linear programming problems involving two variables. These problems are amenable to geometric analysis, and the method of solution introduced here will shed much light on the basic nature of a linear programming problem.
- 3. Graphing Linear Inequalities • A linear equation in two variables x and y, has a solution set that may be exhibited graphically as points on a straight line in the xy-plane. • There is also a simple graphical representation for linear inequalities in two variables: • Before turning to a general procedure for graphing such inequalities, let’s consider a specific example. Suppose we wish to graph • We first graph the equation 2x 3y = 6, which is obtained by replacing the given inequality “<” with an equality “=”.
- 7. Procedure for Graphing Linear Inequalities: 1. Draw the graph of the equation obtained for the given inequality by replacing the inequality sign with an equal sign. Use a dashed or dotted line if the problem involves a strict inequality, < or >. Otherwise, use a solid line to indicate that the line itself constitutes part of the solution. 2. Pick a test point (a, b) lying in one of the half-planes determined by the line sketched in Step 1 and substitute the numbers a and b for the values of x and y in the given inequality. For simplicity, use the origin whenever possible. 3. If the inequality is satisfied, the graph of the solution to the inequality is the half-plane containing the test point. Otherwise, the solution is the half-plane not containing the test point.
- 22. Reference Finite Mathematics for the Managerial, Life, and Social Sciences, Tenth Edition