Systems of Equations and Inequalities

1. Systems of Equations and Inequalities Lesson 6.2

2. Recall … Number of Solutions • System of linear equations • One solution • System is consistent … equations are independent • No solutions • System is inconsistent … equations still independent • Many (infinite) solutions • System is consistent … equations are dependent

3. "Elimination" Solution Method • Given system • Eliminate one of the variables by adding the two equations together • Then solve for remaining variable • Now substitute result back into one of equations to determine 2nd variable

4. "Elimination" Solution Method • Note results of this method when system is inconsistent or dependent • Try these … Can you come up with a "rule of thumb" which tells you when a system is either inconsistent or dependent? Hint … multiply both sides of bottom equation by some constant

5. Systems of Inequalities • Linear inequality in two variables written as a x + b y ≤ c • Note ≤ could also be <, >, or ≥ • Graph of a linear inequality is a "half plane" • Represents all ordered pairs which satisfy the inequality

6. Note: ≤ or ≥ means that line of equation is included – graph as solid. • Otherwise line is dotted Example • Given 2x + 3y ≤ 6 • Solve for y y ≤ -2/3x +2 • Graph equation • Choose orderedpair from one sideor 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

7. Systems of Inequalities • We seek the ordered pairs which satisfy all inequalities • Try this system

8. What is this point? Application • A rectangular pen forSnidly's pet monster is tobe made out of 40 ft of fence • Let y = length, x = width • We know • Which sides of thelines are included?

9. Application • What dimensions give an area of 91 ft2 ? 91 y x

10. Application • What is the formula for A in terms of y? • Graph A • What is the maximum area possible for the pen?

11. Assignment A • Lesson 6.2A • Page 477 • Exercises 1 – 67 odd

12. 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

13. Linear Programming Example • Company produces 2 products • CD players • Radios • Constraints • Must produce 5 ≤ radios ≤ 25 • Radios produced ≤ CD players produced • CD players produced ≤ 30 • Profit • $35 per CD player • $15 per radio What linear inequalities are expressed by these constraints? We need a linear objective function – what is a function which gives profit?

14. Linear Programming Example • Let radios be x, CD players be y • Profit = 15x + 35y • Constraints • x ≥ 5 • x ≤ 25 • x ≤ y • y ≤ 30 • Now determinevertices of region (25,30) (5,30) (25,25) (5,5)

15. 250 1250 1425 1125 Linear Programming Example • Next plug those vertex ordered pairs into the profit function • Vertex with largest value will be combination to use

16. Fundamental Theorem of Linear Programming • If the optimal value exists • It will occur at a vertex of the region of feasible solutions

17. 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)

18. 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?

19. Assignment B • Lesson 6.2B • Page 480 • Exercises 75 – 91 odd