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1. The Linear Equations Chapter The context of the Linear Equations Chapter is Choosing a Cellphone Plan. The goal is to compare several different plans to find the least expensive plan, but with the features needed. The rst provider is Metro PCS, which offers unlimited minutes per month for a Flat fee of 9660. The second provider is Verizon, which costs $39.99 per month for 450 minutes and 50.45 for each additional minute. The comparative analysis will be made just on these considerations, so let a variable x be the number of minutes that you talk on your cell phone each month. (a) Suppose that M is the cost per month for the Metro PCS Plan. Find a linear equation involving M to represent the amount that you will have to pay. (b) Now let V be the cost per month for the Verizon Plan. Find a linear equation involving V and x, where 0 S x S 450. Also find a linear equation involving V and x, where x > 450. (c) On the supplied graph paper, graph the three linear equations from (a) and (b) on a single coordinate system, for 0 S x S 500 (d) Find the number of minutes at which the Metro PCS plan becomes the cheaper plan each month by nding the point 2 of intersection of two of the graphs. Label the point on the graph, and round to the nearest minute. (e) Explain how you can use the solution in {dl to see which plan is more economical. The Systems of Linear Equations Chapter The two plans will now be further analyzed, taking into consideration the costs for text messages. The Metro PCS Plan charges $0.04 per text message and the Verizon Plan charges $0.02 per text message. The charges for minutes used (variable x) are as in 1. Let the variable y represent the number of text messages used per month. (a) For x 2 0, express the cost M as a linear expression in the variables x and y, by including the text message charges with 1(a} (b) For x 2 450, express the cost V as a linear expression in the variables x and y, by including the text message charges with 1{b) (c) If 500 minutes are used for a month, find the value ofy so that M = V and round up. (d) If 350 text messages are used in a month, and x 2 450, nd the value of x so that M S V 3. The Linear Programming Chapter The goal in the Linear Programming Chapter is to decide how to make a trail mix from peanuts and raisins. The following table shows the number of carbohydrates in one cup of peanuts and one cup of raisins. Carboh drates (rams -er cu-) 31.40 114.74 The desired amounts should satisfy these conditions. (i) There are at most 6 cups of the mix (ii) The amount in cups of raisins should be at least 1/2 ofthe amount of peanuts (iii) The amount in cups of peanuts should be at least 1/2 of the amount of raisins (iv) There should be at least one cup of peanuts The goal is to maximize the amount of carbohydrates in the mix (a) Let x be the number of cups of peanuts in the mix, and let y be the number of cups of raisins in the mix. Express each of the requirements {i}, (ii), and (iii) above in terms of inequalities in the variables x and y (b) Set c be the number of carbohydrates in the mix, and express c as a linear expression in the variables x and y, using the above values in the table (c) Graph the inequalities in {a) on the supplied graph paper on a single coordinate system and show the solution region, with x 2 0 and y 2 0 (d) Identify the corner points of the solution region (e) Evaluate c at each corner point to find the number of cups of peanuts and the number of cups of raisins that maximizes the number of carbohydrates per cup in the mix (f) How many grams of carbohydrates are in one cup of the final mix