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All Textbook Solutions for Precalculus
For Exercises 63-64, determine the number of solutions to the system of equations. y=2x+11+log2y=xFor Exercises 63-64, determine the number of solutions to the system of equations. x2y2=0x=yFor Exercises 65-70, solve the system. logx+2logy=52logxlogy=0For Exercises 65-70, solve the system. log2x+3log2y=6log2xlog2y=2For Exercises 65-70, solve the system. 2x+2y=64x2y=14For Exercises 65-70, solve the system. 3x9y=183x+3y=30For Exercises 65-70, solve the system. x12+y+12=5x2+y+42=29For Exercises 65-70, solve the system. x+32+y22=4x12+y2=8For Exercises 71-72, use substitution to solve the system for the set of ordered triples x,y, that satisfy the system. 2=2x6=2yx2+y2=10For Exercises 71-72, use substitution to solve the system for the set of ordered triples x,y, that satisfy the system. 8=4x2=2y2x2+y2=973PEThe minimum and maximum distances from a point P to a circle are found using the line determined by the given point and the center of the circle. Given the circle defined by x2+y2=9 and the point P4,5, a. Find the point on the circle closest to the point 4,5. b. Find the point on the circle farthest from the point 4,5.For Exercises 75-80, use a graphing utility to approximate the solution (s) to the system of equations. Round the coordinates to 3 decimal places. y=0.6x+7y=ex5For Exercises 75-80, use a graphing utility to approximate the solution (s) to the system of equations. Round the coordinates to 3 decimal places. y=0.7x+4y=lnxFor Exercises 75-80, use a graphing utility to approximate the solution (s) to the system of equations. Round the coordinates to 3 decimal places. x2+y2=40y=x2+8.5For Exercises 75-80, use a graphing utility to approximate the solution (s) to the system of equations. Round the coordinates to 3 decimal places. x2+y2=32y=0.8x29.2For Exercises 75-80, use a graphing utility to approximate the solution (s) to the system of equations. Round the coordinates to 3 decimal places. y=x28x+20y=4logxFor Exercises 75-80, use a graphing utility to approximate the solution (s) to the system of equations. Round the coordinates to 3 decimal places. y=0.2exy=0.6x22x3Graph the solution set. 4xy3Graph the solution set. 2y5xGraph the solution set. a. 2y6 b. x24Graph the solution set. x2+y+1216Graph the solution set to the system of inequalities. y13x+12x+y16SPA family plans to spend two 8-hr days at Disney World and will split time between the Magic Kingdom and Epcot Center. Let x represent the number of hours spent at the Magic Kingdom and let y represent the number of hours spent at Epcot Center. a. Set up two inequalities that indicate that the number of hours spent at the Magic Kingdom and the number of hours spent at Epcot cannot be negative. b. Set up an inequality that indicates that the combined number of hours spent at the two parks is at most 16hr . c. Graph the solution set to the system of inequalities.An inequality that can be written in the form Ax+ByC (where A and B are not both zero) is called a inequality in two variables.For a constant real number k, the inequality xk represents the half-plane to the (left/right) of the (horizontal/vertical) line x=k.For a constant real number k, the inequality yk represents the half-plane (above/below) the (horizontal/vertical) line y=k.Given the inequality y2x+1, the bounding line y=2x+1 is drawn as a (dashed/solid) line.The solution set to the system of inequalities x0,y0 represents the points in quadrant I,II,III,IV .The equation x2+y2=4 is a circle centered at with radius . The solution set to the inequality x2+y24 represents the set of points (inside/outside) the circle x2+y2=4.For Exercises 7-10, determine whether the ordered pair is a solution to the inequality. 3x+4y12 a. 1,3 b. 5,1 c. 4,0For Exercises 7-10, determine whether the ordered pair is a solution to the inequality. 2x+3y6 a. 3,3 b. 5,1 c. 0,2For Exercises 7-10, determine whether the ordered pair is a solution to the inequality. yx32 a. 3,30 b. 1,4 c. 5,5For Exercises 7-10, determine whether the ordered pair is a solution to the inequality. yx31 a. 1,2 b. 2,6 c. 4,50a. Graph the solution set. 4x5y20 (See Example 1) b. Explain how the graph would differ for the inequality 4x5y20. c. Explain how the graph would differ for the inequality 4x5y20.a. Graph the solution set. 2x+5y10 b. Explain how the graph would differ for the inequality 2x+5y10. c. Explain how the graph would differ for the inequality 2x+5y10.For Exercises 13-24, graph the solution set. (See Examples 1-3) 2x+5y5For Exercises 13-24, graph the solution set. (See Examples 1-3) 5x+4y8For Exercises 13-24, graph the solution set. (See Examples 1-3) 30x20y+600For Exercises 13-24, graph the solution set. (See Examples 1-3) 400x100y+8000For Exercises 13-24, graph the solution set. (See Examples 1-3) 5x6yFor Exercises 13-24, graph the solution set. (See Examples 1-3) 3x2yFor Exercises 13-24, graph the solution set. (See Examples 1-3) 3+2x+yy+3For Exercises 13-24, graph the solution set. (See Examples 1-3) 43xy2y4For Exercises 13-24, graph the solution set. (See Examples 1-3) x6For Exercises 13-24, graph the solution set. (See Examples 1-3) y5For Exercises 13-24, graph the solution set. (See Examples 1-3) 12y+45For Exercises 13-24, graph the solution set. (See Examples 1-3) 13x+24a. Graph the solution set. x2+y24 (See Example 4) b. Explain how the graph would differ for the inequality x2+y24. c. Explain how the graph would differ for the inequality x2y24.a. Graph the solution set yx21 b. Explain how the graph would differ for the inequality yx21. c. Explain how the graph would differ for the inequality yx21.For Exercises 27-36, graph the solution set. (See Example 4) yx2For Exercises 27-36, graph the solution set. (See Example 4) x2+y216For Exercises 27-36, graph the solution set. (See Example 4) yx22+1For Exercises 27-36, graph the solution set. (See Example 4) yx+122For Exercises 27-36, graph the solution set. (See Example 4) x3For Exercises 27-36, graph the solution set. (See Example 4) y2For Exercises 27-36, graph the solution set. (See Example 4) 2y2For Exercises 27-36, graph the solution set. (See Example 4) x+13For Exercises 27-36, graph the solution set. (See Example 4) yxFor Exercises 27-36, graph the solution set. (See Example 4) yx1a. Is the point 2,1 a solution to the inequality y2x+3? b. Is the point 2,1 a solution to the inequality x+y1? c. Is the point 2,1 a solution to the system of inequalities? y2x+3x+y1a. Is the point 3,2 a solution to the inequality yx+5? b. Is the point 3,2 a solution to the inequality 3x+y11? c. Is the point 3,2 a solution to the system of inequalities? yx+53x+y11For Exercises 39-40, determine whether the ordered pair is a solution to the system of inequalities. x+y4y2x+1 a. 0,1 b. 3,1 c. 2,0 d. 1,4For Exercises 39-40, determine whether the ordered pair is a solution to the system of inequalities. xx2+3x+2y2 a. 2,1 b. 0,2 c. 0,1 d. 3,6For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x12x4y2x+1For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) y13x2yx4For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) 2x+5y53x+4y4For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) 4x3y3x+4y4For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x2+y29x2+y216For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x2+y21x2+y225For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) y3x+33x+y1For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) y2x42x+y2For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x3y3For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x2y2For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) yx22yxy4For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) yx2+7yx+5y1For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x2+y2100y43xx8For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x2+y2100yxy1For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) yexy1x2For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) y2xy0yxFor Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x+22+y329xy2For Exercises 41-58, graph the solution set. If there is no solution, indicate that the solution set is the empty set. (See Examples 5-6) x42+y+12252xy4For Exercises 59-64, write an inequality to represent the statement. x is at most 6.For Exercises 59-64, write an inequality to represent the statement. y is no more than 7.For Exercises 59-64, write an inequality to represent the statement. y is at least 2.For Exercises 59-64, write an inequality to represent the statement. x is no less than 12.For Exercises 59-64, write an inequality to represent the statement. The sum of x and y does not exceed 18.For Exercises 59-64, write an inequality to represent the statement. The difference of x and y is not less than 4.Let x represent the number of hours that Trenton spends studying algebra, and let y represent the number of hours he spends studying history. For parts (a)-(e), write an inequality to represent the given statement. (See Example 7) a. Trenton has a total of at most 9hr to study for both algebra and history combined. b. Trenton will spend at least 3hr studying algebra. c. Trenton will spend no more than 4hr studying history. d. The number of hours spent studying algebra cannot be negative. e. The number of hours spent studying history cannot be negative. f. Graph the solution set to the system of inequalities from parts (a)-(e)Let x represent the number of country songs that Sierra puts on a playlist on her portable media player. Let y represent the number of rock songs that she puts on the playlist. For parts (a)-(e), write an inequality to represent the given statement. a. Sierra will put at least 6 country songs on the playlist. b. Sierra will put no more than 10 rock songs on the playlist. c. Sierra wants to limit the length of the playlist to at most 20 songs. d. The number of country songs cannot be negative. e. The number of rock songs cannot be negative. f. Graph the solution set to the system of inequalities from parts (a)-(e).A couple has $60,000 to invest for retirement. They plan to put x dollars in stocks and y dollars in bonds. For parts (a)-(d), write an inequality to represent the given statement. a. The total amount invested is at most $60,000. b. The couple considers stocks a riskier investment, so they want to invest at least twice as much in bonds as in stocks. c. The amount invested in stocks cannot be negative. d. The amount invested in bonds cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).A college theater has a seating capacity of 2000. It reserves x tickets for students and y tickets for general admission. For parts (a)-(d) write an inequality to represent the given statement. a. The total number of seats available is at most 2000. b. The college wants to reserve at least 3 times as many student tickets as general admission tickets. c. The number of student tickets cannot be negative. d. The number of general admission tickets cannot be negative. e. Graph the solution set to the system of inequalities from parts (a)-(d).Write a system of inequalities that represents the points in the first quadrant less than 3 units from the origin.Write a system of inequalities that represents the points in the second quadrant more than 4 units from the origin.Write a system of inequalities that represents the points inside the triangle with vertices 3,4,3,2, and 5,4.Write a system of inequalities that represents the points inside the triangle with vertices 4,4,1,1, and 5,1.A weak earthquake occurred roughly 9km south and 12km west of the center of Hawthorne, Nevada. The quake could be felt 16km away. Suppose that the origin of a map is placed at the center of Hawthorne with the positive x-axis pointing east and the positive y-axis pointing north. a. Find an inequality that describes the points on the map for which the earthquake could be felt. b. Could the earthquake be felt at the center of Hawthorne?A coordinate system is placed at the center of a town with the positive x-axis pointing east, and the positive y-axis pointing north. A cell tower is located 4mi west and 5mi north of the origin. a. If the tower has a 8-mi range, write an inequality that represents the points on the map serviced by this tower. b. Can a resident 5mi east of the center of town get a signal from this tower?Under what circumstances should a dashed line or curve be used when graphing the solution set to an inequality in two variables?Explain how test points are used to determine the region of the plane that represents the solution to an inequality in two variables.Explain how to find the solution set to a system of inequalities in two variables.Describe the solution set to the system of inequalities. x0,y0,x1,y179PE80PEFor Exercises 81-82, use a graphing utility to graph the solution set to the system of inequalities. y0.4exy0.25x34xFor Exercises 81-82, use a graphing utility to graph the solution set to the system of inequalities. y4x2+1y2x2+0.5For Exercises 1-2, for parts (a) and (b), graph the equation. For part (c), solve the system of equations. For parts (d) and (e) graph the solution set to the system of inequalities. If there is no solution, indicate that the solution set is the empty set. a. y=3x+5 b. 2x+y=0 c. y=3x+52x+y=0 d. y3x+52x+y0 e. y3x+52x+y0For Exercises 1-2, for parts (a) and (b), graph the equation. For part (c), solve the system of equations. For parts (d) and (e) graph the solution set to the system of inequalities. If there is no solution, indicate that the solution set is the empty set. a.y=2x3b.4x2y=2y=2x3c.4x2y=2y2x3d.4x2y2y2x3e.4x2y2For Exercises 3-4, for part (a), graph the equations in the system and determine the solution set. For parts (b) and (c), graph the solution set to the inequality. a. y=x2y=12x2 b. yx2y12x2 c. yx2y12x2For Exercises 3-4, for part (a), graph the equations in the system and determine the solution set. For parts (b) and (c), graph the solution set to the inequality. a. xy=1y=x32 b. xy1yx32 c. xy1yx32An office manager needs to staff the office. She hires full-time employees at $36 per hour and part-time employees at $24 per hour. Write an objective function that represents the total cost in$ to staff the office with x full-time employees and y part-time employees for 1hr.2SP3SPA manufacturer produces two sizes of leather handbags. It takes longer to cut and dye the leather for the smaller bag, but it takes more time sewing the larger bag. The production constraints and profit for each type of bag are given in the table. The machinery limits the number of bags produced to at most 1000 per week. If the company has 900hr per week available for cutting and dying and 1800hr available per week for sewing, determine the number of each type of bag that should be produced weekly to maximize profit. Assume that all bags produced are also sold.The process that maximizes or minimizes a function subject to linear constraints is called programming.The function to be optimized in a linear programming application is called the function.The region in the plane that represents the solution set to a system of constraints is called the region.The points of intersection of a feasible region are called the of the region.A diner makes a profit of $0.80 for a cup of coffee and $1.10 for a cup of tea. Write an objective function z=fx,y that represents the total profit for selling x cups of coffee and y cups of tea. (See Example 1)Rita burns 10 calories per minute running and 8 calories per minute lifting weights. Write an objective function z=fx,y that represents the total number of calories burned by running for x minutes and lifting weights for y minutes.A courier company makes deliveries with two different trucks. Truck A costs $0.62/mi to operate and truck B costs 0.50/mi to operate. Write an objective function z=fx,y that represents the total cost for driving truck A for x miles and driving truck B for y miles.The cost for an animal shelter to spay a female cat is $82 and the cost to neuter a male cat is $55. Write an objective function z=fx,y that represents the total cost for spaying x female cats and neutering y male cats.For Exercises 9-12, a. Determine the values of x and y that produce the maximum or minimum value of the objective function on the given feasible region. b. Determine the maximum or minimum value of the objective function on the given feasible region. Maximize: z=3x+2yFor Exercises 9-12, a. Determine the values of x and y that produce the maximum or minimum value of the objective function on the given feasible region. b. Determine the maximum or minimum value of the objective function on the given feasible region. Maximize: z=1.8x+2.2yFor Exercises 9-12, a. Determine the values of x and y that produce the maximum or minimum value of the objective function on the given feasible region. b. Determine the maximum or minimum value of the objective function on the given feasible region. Minimize: z=1000x+900yFor Exercises 9-12, a. Determine the values of x and y that produce the maximum or minimum value of the objective function on the given feasible region. b. Determine the maximum or minimum value of the objective function on the given feasible region. Minimize: z=6x+9yFor Exercises 13-18, a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of x and y that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. x0,y0x+y60y2x Maximize: z=250x+150yFor Exercises 13-18, a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of x and y that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. x0,y02x+y40x+2y50 Maximize: z=9.2x+8.1yFor Exercises 13-18, a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of x and y that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. x0,y03x+y502y+y40 Maximize: z=3x+2yFor Exercises 13-18, a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of x and y that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. x0,y04x+3y602x+3y36Maximize:z=4.5x+6yFor Exercises 13-18, a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of x and y that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. x0,y0x36y40x+y48 Maximize: z=150x+90yFor Exercises 13-18, a. For the given constraints, graph the feasible region and identify the vertices. b. Determine the values of x and y that produce the maximum or minimum value of the objective function on the feasible region. c. Determine the maximum or minimum value of the objective function on the feasible region. x0,y0410y8x+y12 Maximize: z=50x+70yFor Exercises 19-20, use the given constraints to find the maximum value of the objective function and the ordered pair x,y that produces the maximum value. x0,y03x+4y482x+y22y9 a. Maximize: z=100x+120y b. Maximize: z=100x+140yFor Exercises 19-20, use the given constraints to find the maximum value of the objective function and the ordered pair x,y that produces the maximum value. x0,y0x+y20x+2y36y14 a. Maximize: z=12x+15y b. Maximize: z=15x+12yA furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is $240 and the profit is 160. The cost to build a dining room table is $320 and the profit is $240. (See Examples 2-3) Let x represent the number of kitchen tables produced per month. Let y represent the number of dining room tables produced per month. a. Write an objective function representing the monthly profit for producing and selling x kitchen tables and y dining room tables. b. The manufacturing process is subject to the following constraints. Write a system of inequalities representing the constraints. • The number of each type of table cannot be negative. • Due to labor and equipment restrictions, the company can build at most 120 kitchen tables. • The company can build at most 90 dining room tables. • The company does not want to exceed a monthly cost of $48,000. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.) g. What is the maximum profit?Guyton makes $24/hr tutoring chemistry and $20/hr tutoring math. Let x represent the number of hours per week he spends tutoring chemistry. Let y represent the number of hours per week he spends tutoring math. a. Write an objective function representing his weekly income for tutoring x hours of chemistry and y hours of math. b. The time that Guyton devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints. • The number of hours spent tutoring each subject cannot be negative. • Due to the academic demands of his own classes he tutors at most 18hr per week. • The tutoring center requires that he tutors math at least 4hr per week. • The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex f. How many hours tutoring math and how many hours tutoring chemistry should Guyton work to maximize his income? g. What is the maximum income? h. Explain why Guyton's maximum income is found at a point on the line x+y=18.A plant nursery sells two sizes of oak trees to landscapers. Large trees cost the nursery $120 from the grower. Small trees cost the nursery $80. The profit for each large tree sold is $35 and the profit for each small tree sold is $30. The monthly demand is at most 400 oak trees. Furthermore, the nursery does not want to allocate more than $43,200 each month on inventory for oak trees. a. Determine the number of large oak trees and the number of small oak trees that the nursery should have in its inventory each month to maximize profit. (Assume that all trees in inventory are sold.) b. What is the maximum profit? c. If the profit on large trees were $50, and the profit on small trees remained the same, then how many of each should the nursery have to maximize profit?A sporting goods store sells two types of exercise bikes. The deluxe model costs the store $400 from the manufacturer and the standard model costs the store $300 from the manufacturer. The profit that the store makes on the deluxe model is 180 and the profit on the standard model is 120 . The monthly demand for exercise bikes is at most 30 . Furthermore, the store manager does not want to spend more than $9600 on inventory for exercise bikes. a. Determine the number of deluxe models and the number of standard models that the store should have in its inventory each month to maximize profit. (Assume that all exercise bikes in inventory are sold.) b. What is the maximum profit? c. If the profit on the deluxe bikes were $150 and the profit on the standard bikes remained the same, how many of each should the store have to maximize profit?A paving company delivers gravel for a road construction project. The company has a large truck and a small truck. The large truck has a greater capacity, but costs more for fuel to operate. The load capacity and cost to operate each truck per load are given in the table. The company must deliver at least 288yd3 of gravel to stay on schedule. Furthermore, the large truck takes longer to load and cannot make as many trips as the small truck. As a result, the number of trips made by the large truck is at most 34 times the number of trips made by the small truck. a. Determine the number of trips that should be made by the large truck and the number of trips that should be made by the small truck to minimize cost. b. What is the minimum cost to deliver gravel under these constraints?A large department store needs at least 3600 labor hours covered per week. It employs full-time staff 40hr/wk and part-time staff 25hr/wk. The cost to employ a full-time staff member is more because the company pays benefits such as health care and life insurance. The store manager also knows that to make the store run efficiently, the number of full-time employees must be at least 1.25 times the number of part-time employees. a. Determine the number of full-time employees and the number of part-time employees that should be used to minimize the weekly labor cost. b. What is the minimum weekly cost to staff the store under these constraints?A manufacturer produces two models of a gas grill. Grill A requires 1hr for assembly and 0.4hr for packaging. Grill B requires 1.2hr for assembly and 0.6hr for packaging. The production information and profit for each grill are given in the table. (See Example 4) The manufacturer has 1200hr of labor available for assembly and 540hr of labor available for packaging. a. Determine the number of grill A units and the number of grill B units that should be produced to maximize profit assuming that all grills will be sold. b. What is the maximum profit under these constraints? c. If the profit on grill A units is $110 and the profit on grill B units is unchanged, how many of each type of grill unit should the manufacture produce to maximize profit?A manufacturer produces two models of patio furniture. Model A requires 2hr for assembly and 1.2hr for painting. Model B requires 3hr for assembly and 1.5hr for painting. The production information and profit for selling each model are given in the table. The manufacturer has 1200hr of labor available for assembly and 660hr of labor available for painting. a. Determine the number of model A units and the number of model B units that should be produced to maximize profit assuming that all furniture will be sold. b. What is the maximum profit under these constraints? c. If the profit on model A units is $180 and the profit on model B units remains the same, how many of each type should the manufacturer produce to maximize profit?A farmer has 1200 acres of land and plans to plant corn and soybeans. The input cost (cost of seed, fertilizer, herbicide, and insecticide) for 1 acre for each crop is given in the table along with the cost of machinery and labor. The profit for 1 acre of each crop is given in the last column. Suppose the farmer has budgeted a maximum of $198,000 for input costs and a maximum of $110,000 for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre were reversed between the two crops (that is, $100 per acre for corn and $120 per acre for soybeans), how many acres of each crop should be planted to maximize profit?To protect soil from erosion, some farmers plant winter cover crops such as winter wheat and rye. In addition to conserving soil, cover crops often increase crop yields in the row crops that follow in spring and summer. Suppose that a farmer has 800 acres of land and plans to plant winter wheat and rye. The input cost for 1 acre for each crop is given in the table along with the cost for machinery and labor. The profit for 1 acre of each crop is given in the last column. Suppose the farmer has budgeted a maximum of $90,000 for input costs and a maximum of $36,000 for labor and machinery. a. Determine the number of acres of each crop that the farmer should plant to maximize profit. (Assume that all crops will be sold.) b. What is the maximum profit? c. If the profit per acre for wheat were $40 and the profit per acre for rye were $45, how many acres of each crop should be planted to maximize profit?What is the purpose of linear programming?What is an objective function?How is the feasible region determined?If an optimal value exists for an objective function, it exists at one of the vertices of the feasible region. Explain how to find the vertices.Write a system of linear equations represented by the augmented matrix. Then write the solution set. 1231014110012For Exercises 2-4, perform the elementary row operations on the matrix. 231564 R1R2For Exercises 2-4, perform the elementary row operations on the matrix. 231564 12R1R1For Exercises 2-4, perform the elementary row operations on the matrix. 231564 2R1+R2R3For Exercises 5-8, solve the system by using Gaussian elimination or Gauss-Jordan elimination. 2x+y=16x2y=17For Exercises 5-8, solve the system by using Gaussian elimination or Gauss-Jordan elimination. 2x6y=3647y=7x141For Exercises 5-8, solve the system by using Gaussian elimination or Gauss-Jordan elimination. 2x5y+18z=44x3y+11z=27x2y+11z=29For Exercises 5-8, solve the system by using Gaussian elimination or Gauss-Jordan elimination. w+x2z=32x3z=32w+y+z=34yz=9Lily borrowed a total of $10,000 . She borrowed part of the money from her friend Sly who did not charge her interest. She borrowed part of the money from a credit union at 5 simple interest, and she borrowed the rest of the money from a bank at 7.5 interest At the end of 1 yr, she owed 500 in interest If she borrowed $1000 less from her friend than she did from the bank, determine how much she borrowed from each source.For Exercises 10-13, determine the solution set for the system represented by each augmented matrix. 104010For Exercises 10-13, determine the solution set for the system represented by each augmented matrix. 126001For Exercises 10-13, determine the solution set for the system represented by each augmented matrix. 140000For Exercises 10-13, determine the solution set for the system represented by each augmented matrix. 103001210000For Exercises 14-19, solve the system by using Gaussian elimination or Gauss-Jordan elimination. 3x+6y=9x+2y=3For Exercises 14-19, solve the system by using Gaussian elimination or Gauss-Jordan elimination. 2xy=8yy=x6For Exercises 14-19, solve the system by using Gaussian elimination or Gauss-Jordan elimination. x2y=3z10xy=z73x7y11z=320For Exercises 14-19, solve the system by using Gaussian elimination or Gauss-Jordan elimination. x3z=52x+y+10z=7x+y+z=8For Exercises 14-19, solve the system by using Gaussian elimination or Gauss-Jordan elimination. 2x=3yz4x2y=z+2y2x+y=2z2For Exercises 14-19, solve the system by using Gaussian elimination or Gauss-Jordan elimination. 5y=x+2z+12x5y+4z=23x+2z=15y3The solution set to a system of dependent equations is given. Write three ordered triples that are solutions to the system. Answers may vary. 2z3,z+2,zzisanyrealnumbera. Assume that traffic flows freely around the traffic circle. The flow rates given are measured in vehicles per hour. If the flow rate x3 is 130 vehicles per hour, determine the flow rates x1 and x2 . b. If traffic between intersections B and C flows at a rate of between 100 and 150 vehicles per hour, inclusive, find the range of values for x1 and x2 .a. Assume that traffic flows freely through intersections A,B,C, and D and that all flow rates are measured in vehicles per hour. If the flow rate x4 is 220 vehicles per hour, find the flow rates x1,x2, and x3. b. If the flow rate x4 is between 200 and 250 vehicles per hour, inclusive, find the range of values x1,x2, and x3.For Exercises 23-26. a. Give the order of the matrix. b. Classify the matrix as a square matrix, row matrix, column matrix, or none of these. 1234.12For Exercises 23-26. a. Give the order of the matrix. b. Classify the matrix as a square matrix, row matrix, column matrix, or none of these. 5416For Exercises 23-26. a. Give the order of the matrix. b. Classify the matrix as a square matrix, row matrix, column matrix, or none of these. 3.18.7For Exercises 23-26. a. Give the order of the matrix. b. Classify the matrix as a square matrix, row matrix, column matrix, or none of these. 3800For Exercises 27-28, determine the value of the given element of the matrix A=aij . A=183469 a21For Exercises 27-28, determine the value of the given element of the matrix A=aij . A=183469 a12For what value of x,y , and z will A=B ? A=34xz and B=y468Solve the equation 3X+A=B for X , given that A=2725 and B=5217 .For Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 3AFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 2BFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 A+BFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 B+CFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 2ACFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 4A+3BFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 ABFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 BCFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 ACFor Exercises 31-40, perform the indicated operations if possible. A=416213 B=237156 C=43105 CAFor Exercises 41-44, perform the indicated operations if possible. A=2614B=13C=27 A2For Exercises 41-44, perform the indicated operations if possible. A=2614B=13C=27 ABFor Exercises 41-44, perform the indicated operations if possible. A=2614B=13C=27 BCFor Exercises 41-44, perform the indicated operations if possible. A=2614B=13C=27 CBA company owns two movie theaters in town. The number of popcorns and drinks sold for each theater is given in matrix Q . The price per item is given in matrix P . Find the product QP and interpret the result. PopcornPopcornDrinksDrinkssmalllargesmalllargeQ=386244418216450382476262Theater1Theater2 P=8.506.505.503.50PopcornsmallPopcornlargeDrinkssmallDrinkslargeMatrix M gives the manufacturer price for four models of dining room tables. Matrix P gives the retail price to the customer. WoodMetalM=1050940890800LargeSmall WoodMetalP=1365122211571040LargeSmall a. Compute PM and interpret its meaning. b. if the tax rate in a certain city is 6 , use scalar multiplication to find a matrix F that gives the final price (including sales tax) to the for each model.a. Write a matrix A that represents the coordinates of the vertices of the triangle, Place the x-coordinate of each point in the first row of A and the corresponding y-coordinate in the second row of A . b. Use addition of matrices to shift the triangle 3 units to the left and 1 unit downward, c. Find the product 1001 . A and explain the effect on the graph of the triangle. d. Find the product 1001 . A and explain the effect on the graph of the triangle.Given two nn matrices A and B , what are the criteria for the matrices to be inverses?Determine whether A and B are inverses. A=4132B=2314Determine whether A and B are inverses. A=1321B=15352515For Exercises 51-56, determine the inverse of the given matrix if possible. Otherwise state that the matrix is singular. A=5212For Exercises 51-56, determine the inverse of the given matrix if possible. Otherwise state that the matrix is singular. A=143818116For Exercises 51-56, determine the inverse of the given matrix if possible. Otherwise state that the matrix is singular. A=231624For Exercises 51-56, determine the inverse of the given matrix if possible. Otherwise state that the matrix is singular. A=101153132For Exercises 51-56, determine the inverse of the given matrix if possible. Otherwise state that the matrix is singular. A=54115124431For Exercises 51-56, determine the inverse of the given matrix if possible. Otherwise state that the matrix is singular. A=131712120210Write the system of equations as a matrix equation of the form AX=B , where A is the coefficient matrix, X is the column matrix of variables, and B is the column matrix of constants. 3x+7y=64x+2z=32xy+5z=13For Exercises 58-61, solve the system using the inverse of the coefficient matrix. 14x+38y=4SeeExercise52forA1.18x116y=2For Exercises 58-61, solve the system using the inverse of the coefficient matrix. 5x2y=26SeeExercise51forA1.x+2y=2For Exercises 58-61, solve the system using the inverse of the coefficient matrix. x+z=2SeeExercise54forA1.x+5y3z=6x3y+2z=3For Exercises 58-61, solve the system using the inverse of the coefficient matrix. 5x+4y+z=6SeeExercise55forA1.15x12y4z=214x3yz=5For Exercises 62-65, refer to the matrix A=aij=542106890 . a. Find the minor of the given element. b. Find the cofactor of the given element. a1363RE64RE65RE66RE67RE68RE69RE70RE71RE72RE73RE74RE75RE76RE77RE78RE1TFor Exercises 1-3, perform the elementary row operations on the matrix A=314215310426 . 3R2+R1R13T4T5T6T7T8T9T10T11T12T13T14TFor Exercises 13-16, solve the system by using Gaussian elimination or Gauss-Jordan elimination. x2y=5z+46y+18z=2x83x+8y+20z=1816T17T18T19T20T21T22T23T24T25T26T27T28T29T30T31T32T1CRE2CRE3CRE4CRE5CRE6CRE7CRE8CRE9CRE10CRE11CRE12CRE13CREFor Exercises 13-14, solve the system of equations. 2xy+5z=2x+4z=0x+y=27zDetermine if the graph of the equation x+y2=9 is symmetric with respect to the x-axis , y-axis , origin, or none of these.Determine if the function defined by fx=6x34x is even, odd, or neither.17CRE18CRE19CRE20CRE21CRE22CRE23CRE24CRE25CRE26CRE27CREFind all zeros of fx=2x3x2+18x9.29CRE30CREa. Write the augmented matrix 7x=9+2y2xy=4 b. Write a system of linear equations represented by the augmented matrix 100183Use the matrix from Example 2 to perform the given row operations. a. R2R3 b. 13R3 c. 4R2+R1R1Solve the system by using Gaussian elimination. 2x+7y+z=14x+3yz=2x+7y+12z=45Solve the system by using Gauss-Jordan elimination. x2y=14x7y=1Solve the system by using Gauss-Jordan elimination. 2x+7y+11z=11x+2y+8x=14x+3y+6x=8A rectangular array of elements is called a .Identify the elements on the main diagonal. 4118205110176Explain the meaning of the notation R2R3.Explain the meaning of the notation R2R2.Explain the meaning of the notation 3R1R1.Explain the meaning of the notation R1R3.