Exam 2 Review Math1324. Solve the system of two equations in two variables. 1) 8x + 7y = 36 3x - 4y = -13 A) (1, 5) B) (0, 5) C) No solution D) (1, 4)

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1 Eam Review Math3 Solve the sstem of two equations in two variables. ) + 7 = = -3 (, 5) B) (0, 5) C) No solution D) (, ) ) = + 30 = -, B) No solution 3 C) , for an real number D) 3, 0 Solve the problem b writing and solving a suitable sstem of equations. 3) If 0 pounds of tomatoes and pounds of bananas cost $7 and 0 pounds of tomatoes and 30 pounds of bananas cost $39, what is the price per pound of tomatoes and bananas? tomatoes: $0.70 per pound; bananas: $0.30 per pound B) tomatoes: $0.0 per pound; bananas: $0.50 per pound C) tomatoes: $0.70 per pound; bananas: $0.50 per pound D) tomatoes: $0.50 per pound; bananas: $0.70 per pound Solve the sstem b back substitution. ) + + z = - + 5z = - z = - (, -5, ) B) (,, -5) C) (-,, -5) D) No solution Obtain an equivalent sstem b performing the stated elementar operation on the sstem. 5) Replace the fourth equation b the sum of itself and 3 times the second equation - + 5z - w = - z - w = z + w = z - w = - + 5z - w = - z - w = z + w = z - w = -7 C) - + 5z - w = - z - w = z + w = z + w = 3 B) D) - + 5z - w = - z - w = z + w = z - w = z - w = - 3z - w = z + w = z - w =

2 Write the sstem of equations associated with the augmented matri. Do not solve ) = 0 = -5 z =- B) = -5 = z = 0 C) = = -3 z = 3 D) = - = 3 z = -3 The reduced row echelon form of the augmented matri of a sstem of equations is given. Find the solutions of the sstem. 7) (3,,, -, w) for an real number w B) (3,,, -) C) (3,,, -, ) D) No solution ) / 3, -,, 0 B) No solution C) 3, 0, -, D) 3, w, -, for an real number w Perform row operations on the augmented matri as far as necessar to determine whether the sstem is independent, dependent, or inconsistent. 9) + - z = 3 + z = z = - Independent B) Dependent C) Inconsistent Solve the sstem of equations. If the sstem is dependent, epress solutions in terms of the parameter z. ) + - z = z = z = z 7, 5 z, z for an real number z B) (, 0, 0) 7 C) (, 5, 7) D) No solution

3 Solve the problem b writing and solving a suitable sstem of equations. ) Alan invests a total of $0,500 in three different was. He invests one part in a mutual fund which in the first ear has a return of %. He invests the second part in a government bond at 7% per ear. The third part he puts in the bank at 5% per ear. He invests twice as much in the mutual fund as in the bank. The first ear Alan's investments bring a total return of $5. How much did he invest in each wa? mutual fund: $7000; bond: $,000: bank: $3500 B) mutual fund: $7000; bond: $,000: bank: $3500 C) mutual fund: $00; bond: $,900: bank: $300 D) mutual fund: $700; bond: $90: bank: $300 ) Linda invests $5,000 for one ear. Part is invested at 5%, another part at %, and the rest at %. The total income from all 3 investments is $00. The income from the 5% and % investments is the same as the income from the % investment. Find the amount invested at each rate. $000 at 5%, $,000 at %, $7000 at % B) $,000 at 5%, $5000 at %, $,000 at % C) $,000 at 5%, $,000 at %, $5000 at % D) $5000 at 5%, $,000 at %, $,000 at % 3) A store sells televisions for $30 and video cassette recorders for $70. At the beginning of the week its entire stock is worth $5,30. During the week it sells three quarters of the televisions and one third of the video cassette recorders for a total of $3,3. How man televisions and video cassette recorders did it have in its stock at the beginning of the week? 7 televisions; 93 video cassette recorders B) televisions; 95 video cassette recorders C) 9 televisions; 9 video cassette recorders D) 90 televisions; 9 video cassette recorders Solve the problem. ) What is the size of the matri? B) 3 C) D) 3 5) What is the size of the matri? 3 B) 3 0 C) 3 D) 3 3 Perform the indicated operation. ) Let C = -3 and D = Find C - 3D. B) - - C) - - D) - 3

4 Solve the problem. 7) Barnes and Able sell life, health, and auto insurance. Sales for Ma and June are given in the matrices. M = Life Health Auto 0,000 5, , ,000 Able Barnes J = 70, ,000 0,000 5,000 3,000 Able Barnes Find the matri that would give total sales for the months of Ma and June. B) C) 90,000 5,000 35,000 50,000 5,000 9,000 90,000 5,000 35,000 50,000 5,000 3,000 D),000 0,000,000 90, ,000 50, ,000

5 Graph the feasible region for the sstem of inequalities. ) B) C) D)

6 9) B) C) D)

7 Find the value(s) of the function on the given feasible region. 0) Find the maimum and minimum of z = -. -5, - B) 0, 0 C) -, 0 D) 0, - Use graphical methods to solve the linear programming problem. ) Maimize z = + 7 subject to: Maimum of 5 when = and = B) Maimum of 3 when = and = 3 C) Maimum of 3 when = 3 and = D) Maimum of when = and = 0 7

8 ) Minimize z = + subject to: Minimum of 3 when = and = 0 B) Minimum of when = 3 and = C) Minimum of 0 when = 0 and = 0 D) Minimum of 9 3 when = 3 and = 3 Find the value(s) of the function, subject to the sstem of inequalities. 3) Find the minimum of P = subject to: 0, 0, +. B) C) D) 3 State the linear programming problem in mathematical terms, identifing the objective function and the constraints. ) A firm makes products A and B. Product A takes hours each on machine L and machine M; product B takes hours on L and hours on M. Machine L can be used for hours and M for 7 hours. Profit on product A is $ and $7 on B. Maimize profit. Maimize A + 7B Subject to: A + B A + B 7 A, B 0. C) Maimize A + 7B Subject to: A + B A + B 7 A, B 0. B) Maimize 7A + B Subject to: A + B A + B 7 A, B 0. D) Maimize A + 7B Subject to: A + B A + B 7 A, B 0. The Acme Class Ring Compan designs and sells two tpes of rings: the VIP and the SST. The can produce up to rings each da using up to 0 total man-hours of labor. It takes 3 man-hours to make one VIP ring, versus man-hours to make one SST ring. 5) How man of each tpe of ring should be made dail to maimize the compan's profit, if the profit on a VIP ring is $30 and on an SST ring is $0? 0 VIP and SST B) VIP and SST C) VIP and SST D) VIP and SST

9 Introduce slack variables as necessar and write the initial simple tableau for the problem. ) Maimize z = + subject to: , 0 s s z C) s s z B) s s z D) s s z Find the pivot in the tableau. 7) in row, column 3 B) in row, column C) in row, column D) in row, column Use the indicated entr as the pivot and perform the pivoting once. ) B) C) D) 9

10 Write the basic solution for the simple tableau determined b setting the nonbasic variables equal to 0. 9) 3 5 z =, = 7, 3 =, = 3, 5 = 3, z = 3 B) =, = 0, 3 = 0, = 3, 5 = 0, z = C) = 0, = 7, 3 =, = 0, 5 = 3, z = 3 D) = 0, = 7, 3 =, = 0, 5 = 3, z = 0 Use the simple method to solve the linear programming problem. 30) Maimize z = subject to: with 0, 0, 3 0 Maimum is 0 when = 0, =, 3 = 0 B) Maimum is 5 when = 0, = 9, 3 = 0 C) Maimum is 0 when = 0, = 9, 3 = 5 D) Maimum is when =, = 0, 3 = 0 A baker makes sweet rolls and donuts. A batch of sweet rolls requires 3 lb of flour, dozen eggs, and lb of sugar. A batch of donuts requires 5 lb of flour, 3 dozen eggs, and lb of sugar. Set up an initial simple tableau to maimize profit. 3) The baker has 50 lb of flour, 0 dozen eggs, 700 lb of sugar. The profit on a batch of sweet rolls is $93.00 and on a batch of donuts is $.00. s s s3 s B) s s s3 s C) s s s3 s D) s s s3 s

11 A manufacturing compan wants to maimize profits on products A, B, and C. The profit margin is $3 for A, $ for B, and $5 for C. The production requirements and departmental capacities are as follows: Department Production requirement b product (hours) Departmental capacit (Total hours) A B C Assembling 3 30,000 Painting 3,000 Finishing 3,000 3) What are the coefficients of the objective function?, 3, B),, C) 3,, 5 D), 3, Solve the problem. 33) An agricultural research scientist is developing three new crop growth supplements -- A, B, and C. Each pound of each supplement contains four enzmes -- E, E, E3, and E -- in the amounts (in milligrams) shown in the table. E E E3 E A 3 B 3 C 3 5 The cost of E is $0/mg, the cost of E is $0/mg, the cost of E3 is $/mg, and the cost of E is also $/mg. The growth benefit for crops is epected to be proportional to times the amount of A used, 5 times the amount of B used, and 0 times the amount of C used. However, the total cost of the enzmes used in A, B, and C must be less than $5000 for each treatment. How man pounds each of A, B, and C should be produced to maimize the growth effect? 3.0, 3.0,.0 B) 9.0, 0, 9.5 C) 0, 0, 5 D) 0, 9.3,.0

12 Answer Ke Testname: MATH3 EXAM REVIEW SPRING 0 ) D ) C 3) B ) B 5) B ) D 7) D ) C 9) B ) A ) B ) B 3) D ) A 5) C ) A 7) A ) B 9) B 0) D ) C ) D 3) B ) D 5) A ) D 7) B ) B 9) C 30) A 3) D 3) C 33) D