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6048Systems of Equations and Inequalitiesof packages weighing 1 pound or less and a surcharge foreach additional pound (or fraction thereof). A customer isbilled $27.75 for shipping a 5-pound package and $64.50for shipping a 20-pound package. Find the base price andthe surcharge for each additional pound.70.Delivery Charges.Refer to Problem 69. Federated Ship-ping, a competing overnight delivery service, informs thecustomer in Problem 69 that it would ship the 5-poundpackage for $29.95 and the 20-pound package for $59.20.(A) If Federated Shipping computes its cost in the samemanner as United Express, find the base price and thesurcharge for Federated Shipping.(B)Devise a simple rule that the customer can use tochoose the cheaper of the two services for each pack-age shipped. Justify your answer.71. Resource Allocation.Acoffee manufacturer usesColombian and Brazilian coffee beans to produce twoblends, robust and mild. A pound of the robust blend re-quires 12 ounces of Colombian beans and 4 ounces ofBrazilian beans. A pound of the mild blend requires 6ounces of Colombian beans and 10 ounces of Brazilianbeans. Coffee is shipped in 132-pound burlap bags. Thecompany has 50 bags of Colombian beans and 40 bags ofBrazilian beans on hand. How many pounds of each blendshould 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 discontinueproduction of the mild blend and only produce the ro-bust blend.SECTION 8-2Gauss-Jordan Elimination•Reduced Matrices•Solving Systems by Gauss–Jordan Elimination•ApplicationNow that you have had some experience with row operations on simple augmentedmatrices, 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. Itturns 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 coefficientmatrix for a system of two equations in two variablesa11x1a12x2k1a21x1a22x2k2a11a21a12a22k1k2•Reduced MatricesPossible Solutions to a Linear SystemIt 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 equationsor the number of variables in the system. The terms unique, consistent, incon-sistent, dependent,and independentare used to describe these solutions, just asthey are for systems with two variables.
8-2Gauss–Jordan Elimination605into one of the following simplified forms:Form 1Form 2Form 3(1)where m, n, and pare real numbers, p0. Each of these reduced forms represents