Math 1324 Review 2(answers) x y 2z R R R

Transcription

1 Math Review (answers). Solve the following system using Gauss-Jordan Elimination. y z R R R R R R z 5 y z R R R z t Infinitely many solutions given by y 5t where t is any real number.. Solve the following system using Gauss-Jordan Elimination. y z y z 7 y z R R R 7R R R R R R 0R R R R R The system has no solution.

2 . a) Solve the following system using Gauss-Jordan Elimination. y z R R R R R 5y 0z R R R The system has infinitely many solutions given by z t 5 9t y 5t where t is any real number. b) Find positive integers y,, and z that solve the system. In order for the system to have positive integer solutions t must be a positive integer and 5 9t and 5t must be positive multiples of. t 5 9t 5t So t must equal and we get the solution, y, z. Find the value(s) of C, if possible, so that the system of equations with the given augmented matri has C 9 C 9 C 9 R R R R R 0 C 6 a) eactly one solution b) infinitely many solutions c) no solution C 6 Not possible C 6

3 5. Consider a linear system whose augmented matri is of the form B 0 a) Is it possible for the system to have no solution? Eplain. No. Since the right sides of the equations are all zeros, 0, y 0, z 0 will always be a solution. b) For what value(s) of B will the system have infinitely many solutions? B 0 R R R R R R B 0 R R R B 0 The system will have infinitely many solutions if B. 6. Solve AX B for X if A and 5 B. AX B, so A AX A B, or X A B 5 9 A B 7. Given that A, B 0 5, and 8 C 6, find the matri X that satisfies the equation X B A C. X B A C, so X A C B

4 8. a) Find the inverse of the matri 0 matri 0. by performing Gauss-Jordan on the augmented R R R R R R 0 0 so b) Solve the system y y8 using the inverse matri method. y 8, so y 8, y c) Solve the system y y using the inverse matri method. y, so y 8, y 8 d) Solve the system y y using the inverse matri method. y, so y, y

5 9. The following matri gives the number of direct flights among the four cities A, B, C, and D. Destination A B C D A 0 0 B 0 0 C D 0 a) Find the number of one-stop flights from city A to city C. This would be the, entry of the square of the matri Origin So there is one-stop flight from city A to city C. b) Find the total number of flights from city B to city C that are either direct or one-stop. This would be the, entry of the sum of the matri and its square So there is flight from city B to city C which is either direct or one-stop.

6 c) Find the matri that gives the number of two-stop flights among these cities. This would be the matri raised to the third power The diagram shows the traffic flow at the intersections of four one-way streets. M Street N Street th Street th Street The traffic rates are in cars per hour. In order to have smooth traffic flow, the number of cars entering an intersection must equal the number of cars leaving an intersection. This leads to four equations, one for each intersection:

7 a) Complete the table of intersection equations. Intersection Equation M Street and th Street 00 N Street and th Street 700 N Street and 0 th Street 600 M Street and 0 th Street 000 b) The augmented matri for solving the system of equations using Gauss-Jordan Elimination is Complete Gauss-Jordan Elimination on the augmented matri

8 R R R So the system has infinitely many solutions given by t 600 t 00 t 000 t where t is any real number. c) Since the values of,,, and must be nonnegative, write the four inequalities associated with legitimate solutions, and epress the solutions in terms of a parameter with an inequality. t 0 0 t 600 t 0 00 t t 0 00 t t 600 t t 600 So the solutions of the traffic problem are given by t 600 t 00 t 000 t where t satisfies 0 t 600. d) Determine the maimum and minimum traffic flows on the following street sections: Street Section Minimum Flow Maimum Flow M Street between 0 th and th th between M Street and N Street N Street between 0 th and th th between M Street and N Street 0 600

9 e) If traffic on th between M Street and N Street is restricted to 00 cars per hour due to construction, determine the traffic flow in the rest of the system. 00, which means that t 00, so 800, 00, 00 f) If the following tolls are charged, determine the least and greatest amount of money generated from the tolls. Money t t t 5t 000 0t Street Section Toll Minimum Maimum M Street between 0 th and th $.5 th between M Street and N Street $.50 $0 $50 N Street between 0 th and th $.0 0 th between M Street and N Street $.5. A State Fish and Game Department will supply three types of food to a lake that can support three species of fish. Each fish of Species consumes, each week, an average of unit of Food, unit of Food, and units of Food. Each fish of Species consumes, each week, an average of units of Food, units of Food, and 5 units of Food. For a fish of Species, the average weekly consumption is units of Food, unit of Food, and 5 units of Food. Each week 5,000 units of Food, 0,000 units of Food, and 55,000 units of Food are supplied to the lake. If we asume that all food is eaten, we d like to know how many of each type of fish can coeist in the lake. Food Equation 5,000 Food Equation 0,000 Food Equation ,000 a) Complete the augmented matri for the system of equations. 5,000 0, ,000 b) The result of performing Gauss-Jordan Elimination on the augmented matri is the following:

10 0 5 0, , Since the values of,, and must be nonnegative whole numbers, write the four inequalities associated with legitimate solutions, and epress the solutions in terms of a parameter with an inequality. t 0 0 t t 5, ,000 5t 0 5,000 t t 8,000 5,000 t 8,000 So the solutions are given by 5,000 t 8,000. t t5,000 where t is a whole number that satisfies 0,000 5t c) What is the largest number of fish Species that can survive in the lake?,000 d) What is the smallest number of fish Species that can survive in the lake? 5,000 e) What is the largest total population of all three species that can coeist in the lake? Total t t 5,000 0,000 5t 5,000 t So the largest total population is 0,000. f) What is the smallest total population of all three species that can coeist in the lake? Total t t 5,000 0,000 5t 5,000 t So the smallest total population is,000.