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Systems of Equations and Inequalities Lesson 6.2 Recall … Number of Solutions System of linear equations One solution No solutions System is consistent … equations are independent System is inconsistent … equations still independent Many (infinite) solutions System is consistent … equations are dependent "Elimination" Solution Method Given system 2 x y 15 x y 0 2 x y 15 x y0 3 x 0 y 15 x5 Eliminate one of the variables by adding the two equations together Then solve for remaining variable Now substitute result back into one of equations 5 y 0 to determine 2nd variable 5 y "Elimination" Solution Method Note results of this method when system is inconsistent or dependent Try these … x 3y 1 4 x 2 y 10 2x 6 y 2 2 x y 10 Hint … multiply both sides of bottom equation by some constant Can you come up with a "rule of thumb" which tells you when a system is either inconsistent or dependent? Systems of Inequalities Linear inequality in two variables written as ax+by≤c Note ≤ could also be <, >, or ≥ Graph of a linear inequality is a "half plane" Represents all ordered pairs which satisfy the inequality Example Given 2x + 3y ≤ 6 • Note: ≤ or ≥ means that line of equation is included – graph as solid. • Otherwise line is dotted Solve for y Graph equation y ≤ -2/3x +2 Choose ordered pair from one side or the other (0, 0) is an easy choice Determine if that ordered pair satisfies the inequality If so – that's the side, if not – other side Systems of Inequalities We seek the ordered pairs which satisfy all inequalities Try this system x y 3 x y 3 Application A rectangular pen for Snidly's pet monster is to be made out of 40 ft of fence Let y = length, x = width We know 2 x 2 y 40 and Which sides of the lines are included? What is this point? yx Application What dimensions give an area of 91 ft2 ? 2 x 2 y 40 x y 91 y 91 x Application What is the formula for A in terms of y? 2 x 2 y 40 y 20 x A y x 20 x x Graph A What is the maximum area possible for the pen? Assignment A Lesson 6.2A Page 477 Exercises 1 – 67 odd Linear Programming Procedure used to optimize quantities such as cost and profit Consists of Linear objective function Describes a quantity to be optimized System of linear inequalities called constraints Solution is set of feasible solutions Linear Programming Example Company produces 2 products Constraints CD players Radios What linear inequalities are expressed by these constraints? Must produce 5 ≤ radios ≤ 25 Radios produced ≤ CD players produced CD players produced ≤ 30 Profit $35 per CD player $15 per radio We need a linear objective function – what is a function which gives profit? Linear Programming Example Let radios be x, CD players be y Profit = 15x + 35y Constraints x≥5 x ≤ 25 x≤y y ≤ 30 Now determine vertices of region (5,30) (25,30) (25,25) (5,5) Linear Programming Example Next plug those vertex ordered pairs into the profit function Vertex with largest value will be combination to use Vertex (5,5) (25,25) (25,30) (5,30) P = 15x + 35y 250 1250 1425 1125 Fundamental Theorem of Linear Programming If the optimal value exists It will occur at a vertex of the region of feasible solutions Try It Out For the specified function P = 5x + 3y Find the maximum and minimum values for the region given (2.5, 7) (6.5, 5) (3, 2) (5, 1) Practice We are buying filing cabinets. X costs $100, requires 6 sq ft, holds 8 cu ft Y costs $200, requires 8 sq ft, holds 12 cu ft We can spend a max $1400 We only have 72 sq ft of space We seek maximum storage capacity What are constraints? What is the linear objective function? Graph and solve? Assignment B Lesson 6.2B Page 480 Exercises 75 – 91 odd