Section 3.1 Graphing Systems of Linear Inequalities in Two Variables Procedure for Graphing Linear Inequalities: 1. Draw the graph of the equation obtained for the given inequality by replacing the inequality sign with an equal sign. Use a dashed line if the problem involves a strict inequality, < or >. Otherwise, use a solid line to indicate that the line itself constitutes part of the solution. 2. Pick a test point, (a, b), lying in one of the half-planes determined by the line sketched in Step 1 and substitute the numbers a and b for the values of x and y in the given inequality. For simplicity, use the origin, (0, 0), whenever possible. 3. If the inequality is satisfied (True), the graph of the solution to the inequality is the half-plane containing the test point (Shade the region containing the test point). Otherwise (if the inequality is False), the solution is the half-plane not containing the test point (Shade the region that does not contain the test point). 1. Find the graphical solution of the inequality. 10x +4y apple 8 2. Find the graphical solution of the inequality. 2x +5y> 10
3. Write a system of linear inequalities that describes the shaded region. 5x +2y 30 x +2y 12 4. Determine graphically the solution set for the system of inequalities. Indicate whether the solution set is bounded or unbounded. 3x 2y > 17 x +2y>7 2 Fall 2017, Maya Johnson
5. Determine graphically the solution set for the system of inequalities. Indicate whether the solution set is bounded or unbounded. 3x 4y apple 12 4x +5y apple 20 6. Determine graphically the solution set for the system of inequalities. Indicate whether the solution set is bounded or unbounded. x + y 20 x +2y 30 3 Fall 2017, Maya Johnson
7. Determine graphically the solution set for the system of inequalities. Indicate whether the solution set is bounded or unbounded. x + y apple 5 2x + y apple 8 2x y 1 4 Fall 2017, Maya Johnson
Section 3.3 Graphical Solutions of Linear Programming Problems Theorem 1: Solutions of Linear Programming Problems 1. If a linear programming problem has a solution, then it must occur at a corner point of the feasible set, S, associatedwiththeproblem. 2. If the objective function, P,isoptimizedattwoadjacentcornerpointsofS, thenitisoptimized at every point on the line segment joining the two points (infinitely many solutions). Theorem 2: Existence of a Solution Suppose we are given a linear programming problem with a feasible set S and an objective funtion P = ax + by. 1. If S is bounded then P has both a maximum and a minimum value on S. 2. If S is unbounded and both a and b are nonnegative, then P has a minimum value on S provided that the constraints defining S include the inequalities and. 3. If S is empty, then the linear programming problem has no solution; that is, P has neither a maximum nor a minimum value. We say that the problem is infeasible. The Method of Corners 1. Graph the feasible set. 2. If the feasible set is nonempty, find the coordinates of all corner points of the feasible set. In this class we will use the rref calculator function to find corner points whenever the points are where two lines are crossing. 3. Evaluate the objective function at each corner point. 4. Find the corner point(s) that renders the objective function a maximum (or minimum). 5 Fall 2017, Maya Johnson
1. Find the maximum and/or minimum value(s) of the objective function on the feasible set S. Z =5x +6y 2. Solve the linear programming problem by the method of corners. Maximize P =3x +5y subject to 2x + y apple 16 2x +3y apple 24 y apple 7 6 Fall 2017, Maya Johnson
3. Solve the linear programming problem by the method of corners. Maximize P = x +2y subject to x +2y apple 4 2x +3y 12 7 Fall 2017, Maya Johnson
4. Solve the linear programming problem by the method of corners. Minimize C =3x +6y subject to x +2y 40 x + y 30 8 Fall 2017, Maya Johnson
5. Solve the linear programming problem by the method of corners. Minimize C =2x +4y subject to 4x + y 42 2x + y 30 x +3y 30 9 Fall 2017, Maya Johnson
1 6. Solve the linear programming problem by the method of corners. Find the minimum and maximum of P =7x +2y subject to 3x +5y 20 3x + y apple 16 2x + y apple 2 ( 3,0),( 0,4 ) ( kziol,( 0,16 ) go" " " in' Ye ki F t.be# Eo.EE Test 10,0123 : -6 at o to 42 9 tr on t.ie#e..e. O 42 True - ftp..ee?o Ll EEistfEK@maxsxrsy-zoz.g &L2 ly@7li9isitzl46hheminfg5.p 7.6 7( 2.8 ),f]#fo9 +217.67=348 } +y=16. *i* L2&L3 Bismuth ± e a Di I xt5y=20 2 = - Zxty " ip f*4 HFD 10 Fall 2017, Maya Johnson
7. Perth Mining Company operates two mines for the purpose of extracting gold and silver. The Saddle Mine costs $15, 000/day to operate, and it yields 50 oz of gold and 3000 oz of silver each day. The Horseshoe Mine costs $19, 000/day to operate, and it yields 75 oz of gold and 1000 oz of silver each day. Company management has set a target of at least 650 oz of gold and 18, 000 oz of silver. How many days should each mine be operated so that the target can be met at a minimum cost? (Let x be the # of days Saddle Mine operates and y be the # of days Horseshoe Mine operates.) eraser renren.ee/eeesakneq. \ some 11 Fall 2017, Maya Johnson
8. National Business Machines manufactures x model A fax machines and y model B fax machines. Each model A costs $100 to make, and each model B costs $150. The profits are $40 for each model A and $35 for each model B fax machine. If the total number of fax machines demanded per month does not exceed 2500 and the company has earmarked no more than $600, 000/month for manufacturing costs, how many units of each model should National make each month to maximize its monthly profit? ) ftp.3#*o@4oc25oojt35co)=io@ 0 0 40103+3510=0 n IeFoI:II ind 12 Fall 2017, Maya Johnson