Solving%2BSystem%2Bof%2BLinear%2BEquations%2B-%2BGauss%2BJordan - 604 8 Systems of Equations and Inequalities blends robust and mild A pound of the

# Solving%2BSystem%2Bof%2BLinear%2BEquations%2B-%2BGauss%2BJordan

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604 8 Systems of Equations and Inequalities of packages weighing 1 pound or less and a surcharge for each additional pound (or fraction thereof). A customer is billed \$27.75 for shipping a 5-pound package and \$64.50 for shipping a 20-pound package. Find the base price and the surcharge for each additional pound. 70. Delivery Charges. Refer to Problem 69. Federated Ship- ping, a competing overnight delivery service, informs the customer in Problem 69 that it would ship the 5-pound package for \$29.95 and the 20-pound package for \$59.20. (A) If Federated Shipping computes its cost in the same manner as United Express, find the base price and the surcharge for Federated Shipping. (B) Devise a simple rule that the customer can use to choose the cheaper of the two services for each pack- age shipped. Justify your answer. 71. Resource Allocation. A coffee manufacturer uses Colombian and Brazilian coffee beans to produce two blends, robust and mild. A pound of the robust blend re- quires 12 ounces of Colombian beans and 4 ounces of Brazilian beans. A pound of the mild blend requires 6 ounces of Colombian beans and 10 ounces of Brazilian beans. Coffee is shipped in 132-pound burlap bags. The company has 50 bags of Colombian beans and 40 bags of Brazilian beans on hand. How many pounds of each blend should it produce in order to use all the available beans? 72. Resource Allocation. Refer to Problem 71. (A) If the company decides to discontinue production ofthe robust blend and only produce the mild blend, howmany pounds of the mild blend can it produce and howmany beans of each type will it use? Are there anybeans that are not used? (B) Repeat part A if the company decides to discontinue production of the mild blend and only produce the ro- bust blend. SECTION 8-2 Gauss - Jordan Elimination Reduced Matrices Solving Systems by Gauss Jordan Elimination Application Now that you have had some experience with row operations on simple augmented matrices, we will consider systems involving more than two variables. In addition, we will not require that a system have the same number of equations as variables. It turns out that the results for two-variable two-equation linear systems, stated in The- orem 1 in Section 8-1, actually hold for linear systems of any size. In the last section we used row operations to transform the augmented coefficient matrix for a system of two equations in two variables a 11 x 1 a 12 x 2 k 1 a 21 x 1 a 22 x 2 k 2 a 11 a 21 a 12 a 22 k 1 k 2 Reduced Matrices Possible Solutions to a Linear System It can be shown that any linear system must have exactly one solution, no solu- tion, or an infinite number of solutions, regardless of the number of equations or the number of variables in the system. The terms unique, consistent, incon- sistent, dependent, and independent are used to describe these solutions, just as they are for systems with two variables.
8-2 Gauss–Jordan Elimination 605 into one of the following simplified forms: Form 1 Form 2 Form 3 (1) where m , n , and p are real numbers, p 0. Each of these reduced forms represents