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LrChIIISIVSLE052 - Chapter 3 Lines Parabolas and Systems...

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Chapter 3: Lines, Parabolas, and Systems 3.4: Systems of Linear Equations Dr. Raja Mohammad Latif
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Department of Mathematical Sciences, KFUPM 2 1 3.4: Systems of Linear Equations ===Lecture Sec 3.4 Begins now=== Home Work: 26 ; 28 ; 29 ; 34 ; 37 ; 39 ; 41 Independent System : Lines intersect at one point: exactly one solution. 3 x ° 4 y = 1 ; 2 x + 3 y = 12 : The two lines intersect in exactly one point. S.S. = f (3 ; 2 g : Dependent System : Lines coincide: In°nitely many solutions. ° 4 x + y = 2 ; 8 x ° 2 y = ° 4 : There is an in°nite number of points where the two lines intersect. Inconsistent System : Lines are parallel: no solutions. ° 6 x + 4 y = 7 ; 3 x ° 2 y = 4 : There is no point where the two lines intersect. 177BZ6Example.Diet. A woman wants to use milk and orange juice to increase the amount of calcium and vitamin A in her daily diet. An ounce of milk contains 37 milligrams of calcium and 57 micrograms of vitamin A. An ounce of orange juice contains 5 milligrams of calcium and 65 micrograms of vitamin A. How many ounces of milk and orange juice should the woman drink each day to provide exactly 500 milligrams of calcium and 1200 micrograms of Dr. Raja Latif. Math 131 (052)
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Department of Mathematical Sciences, KFUPM 3 vitamin A? Solution. First, we de°ne the relevant variables: x = Number of ounces of milk y = Number of ounces of orange juice Next, we summarize the given information in the table: Milk Orange Juice Total Needed Calcium 37 5 500 Vitamin A 57 65 1200 Next we use the information in the table to form equations involving x and y 0 @ Calcium in x oz of milk 1 A + 0 @ Calcium in y oz of orang juice 1 A = ° Total calcium needed (mg) ± 37x + 5y = 500 0 @ Vitamin A in x oz of milk 1 A + 0 @ Vitamin A in y oz of orang juice 1 A = ° Total calcium needed (mg) ± 57x + 65y = 1200 Solve using elimination by addition: ° 481x ° 65y = ° 6500 57x + 65y = 1200 ° 424x = ° 5300 = ) x = 12.5 Dr. Raja Latif. Math 131 (052)
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Department of Mathematical Sciences, KFUPM 4 37(12.5) + 5y = 500 = ) 5y =37.5 = ) y = 7.5. Drinking 12.5 ounces of milk and 7.5 ounces of orange juice each day will provide the required amounts of calcium and vitamin A. ========================= 152AL6Example. (Mixture) The Britannia Store, which specializes in selling all kinds of nuts, sells peanuts at $0.70 per pound and cashews at $1.60 per pound. At the end of the month, the owner of the store °nds that peanuts are not selling well and decides to mix peanuts and cashews to make a mixture of 45 pounds, which could sell for $1.00 per pound. How many pounds of peanuts and cashews should be mixed to keep the same revenue? Solution. Let the mixture contain x pounds of peanuts and y pounds of cashews. Since the total mixture is 45 pounds, x + y = 45, The revenue from x pounds of peanuts at $0.70 per pound is 0.70x dollars and the revenue from y pounds of cashews at $1.60 per pound is 1.6y dollars. The revenue obtained from the mixture of 45 pounds at $1.00 per pound will be $45. Dr. Raja Latif. Math 131 (052)
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Department of Mathematical Sciences, KFUPM 5 Since the revenue from the mixture must be the same as from the separate nuts, we have the following equation.
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