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Chapter 4: Problem 51

What is a "basic solution"? How might one find a basic solution of a given system of linear equations?

Short Answer

Expert verified
A basic solution is a solution to a system of linear equations where some variables are assigned the value zero, and others are determined by solving a smaller system. To find a basic solution, first convert the system into an augmented matrix and then into row-echelon form. Identify pivot variables and free variables, express pivot variables in terms of free variables, and create basic solutions by setting free variables to zero.

Step by step solution

01

Rewriting the system as an augmented matrix

Write the given system of linear equations as an augmented matrix. This will make it easier to see the relationships between the variables and the constants in the system. Augmented matrix is obtained by placing the constant terms of the equations to the right of the coefficients matrix, separated by a vertical line.
02

Convert the matrix to row-echelon form

Use elementary row operations (swapping rows, multiplying rows by nonzero constants, or adding/subtracting rows) to convert the augmented matrix to row-echelon form. A matrix is in row-echelon form if each row has at least one nonzero entry, the first nonzero entry in a row (called a pivot) is one, the pivots move strictly to the right as we move down the rows, and rows of zeroes (if any) are at the bottom.
03

Identify pivot variables and free variables

Once the matrix is in row-echelon form, identify the variables corresponding to the pivots (these are called pivot variables) and the variables that don't correspond to pivots (called free variables). The pivot variables are determined by the non-zero elements in each row of the row-echelon form, while the free variables can be assigned any value.
04

Express pivot variables in terms of free variables

For each pivot variable, express it in terms of the free variables by rewriting the corresponding equation in the row-echelon form, isolating the pivot variable on the left side.
05

Create the basic solutions by setting free variables to zero

To find the basic solutions of the system, individually set each of the free variables to zero while keeping the other free variables as parameters. For each setting, compute the values of the pivot variables expressed in terms of these values for the free variables. Each set of values for the pivot variables and the chosen zero-valued free variables forms a basic solution of the system. These are the steps to find the basic solutions of a given system of linear equations. Keep in mind that a basic solution is not necessarily unique in many cases, and not all systems of linear equations will have a basic solution.

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Most popular questions from this chapter

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and 11 grams of carbohydrates. Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 21 grams of carbohydrates. \({ }^{11}\) If the cereal costs \(30 \phi\) per serving and the dessert costs 50 per serving, and you want to provide your child with at least 140 calories and at least 32 grams of carbohydrates, how can you do so at the least cost? (Fractions of servings are permitted.)

Create a linear programming problem in two variables that has more than one optimal solution.

Management \(^{20}\) You are the service manager for a supplier of closed- circuit television systems. Your company can provide up to 160 hours per week of technical service for your customers, although the demand for technical service far exceeds this amount. As a result, you have been asked to develop a model to allocate service technicians' time between new customers (those still covered by service contracts) and old customers (whose service contracts have expired). To ensure that new customers are satisfied with your company's service, the sales department has instituted a policy that at least 100 hours per week be allocated to servicing new customers. At the same time, your superiors have informed you that the company expects your department to generate at least \(\$ 1,200\) per week in revenues. Technical service time for new customers generates an average of \(\$ 10\) per hour (because much of the service is still under warranty) and for old customers generates \(\$ 30\) per hour. How many hours per week should you allocate to each type of customer to generate the most revenue?

Each serving of Gerber Mixed Cereal for Baby contains 60 calories and no vitamin \(C\). Each serving of Gerber Mango Tropical Fruit Dessert contains 80 calories and 45 percent of the U.S. Recommended Daily Allowance (RDA) of vitamin \(\mathrm{C}\) for infants. Each serving of Gerber Apple Banana Juice contains 60 calories and 120 percent of the U.S. RDA of vitamin \(\mathrm{C}\) for infants. \(^{42}\) The cereal costs \(10 \mathrm{~d} /\) serving, the dessert costs \(53 \mathrm{~d} /\) serving, and the juice costs 27 d/serving. If you want to provide your child with at least 120 calories and at least 120 percent of the U.S. RDA of vitamin \(\mathrm{C}\), how can you do so at the least cost? What are your shadow costs for calories and vitamin \(\mathrm{C}\) ?

$$ \begin{array}{ll} \text { Minimize } & c=6 s+6 t \\ \text { subject to } & s+2 t \geq 20 \\ & 2 s+t \geq 20 \\ & s \geq 0, t \geq 0 \end{array} $$
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