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LrChIIISVIApplicationsSystemsEquations052 - 3.6...

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3.6 Applications of Systems of Equations Dr. Raja Mohammad Latif OBJECTIVE: To solve systems describing equilibrium and break-even points.
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Dr. Raja Latif. Math 131 - 052 (Feb. 12 - June 10, 2006) 2 1 3.6 Applications of Systems of Equations ===Lecture Sec 3.6 Begins now=== An equation that relates price per unit and quantity demanded (supplied) is called a demand equation (supply equation). When the demand and supply curves of a product are represented on the same coordinate plane, the point ( q ; p ) = ( m ; n ) where the curves intersect is called the POINT OF EQUILIBRIUM. The price p = n , called the EQUILIBRIUM PRICE, is the price at which consumers will purchase the same quantity of a product that producers wish to sell at that price. The quantity q = m is called the EQUILIBRIUM QUANTITY. Department of Mathematical Sciences, KFUPM
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Dr. Raja Latif. Math 131 - 052 (Feb. 12 - June 10, 2006) 3 In problems 1 ° 8 , you are given a supply equation and a demand equation for a product. If p represents price per unit in dollars and q represents the number of units per unit of time, °nd the equilibrium point. TBQ8. Supply: p = 1 5 q + 5 Demand: p = 3000 q + 20 . [Solution.] Equating p ° values gives 1 5 q + 5 = 3000 q + 20 : Multiplying both sides by 5 ( q + 20 ) gives ( q + 20 )( q + 25 ) = 15000 q 2 + 45q ° 14500 = 0 ( q ° 100 )( q + 145 ) = 0 Therefore q = 100 ; ° 145 : Since q ± 0 ; Choose q = 100 : If q = 100 ; then Department of Mathematical Sciences, KFUPM
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Dr. Raja Latif. Math 131 - 052 (Feb. 12 - June 10, 2006) 4 p = 1 5 q + 5 = 1 5 ( 100 ) + 5 = 25 : The equilibrium point is ( 100 ; 25 ) : ================ Department of Mathematical Sciences, KFUPM
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Dr. Raja Latif. Math 131 - 052 (Feb. 12 - June 10, 2006) 5 In Problems 9 ° 14 ; y TR represents total revenue in dollars and y TC represents total cost in dollars for a manufac- turer. If q represents both the number of units produced and the number of units sold, °nd the break-even quantity. TBQ13. y TR = 100 ° 1000 q + 10 y TC = q + 40 : Solution. Letting y TR = y TC gives 100 ° 1000 q + 10 = q + 40 : Multiplying both sides by q + 10 gives 100 ( q + 10 ) ° 1000 = ( q + 10 )( q + 40 ) q 2 ° 50q + 400 = 0 ( q ° 10 )( q ° 40 ) = 0 Thus q = 10 or 40 units. ===================== Department of Mathematical Sciences, KFUPM
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Dr. Raja Latif. Math 131 - 052 (Feb. 12 - June 10, 2006) 6 TBQ17. Business. A manufacturer sells a product at $ 8 : 35 per unit, selling all products.
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