. So the solution set is parameterized by x = t and y = 2 3 t

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1 Review for Eam II Solutions MAT 2. Solve the following sstem of linear equations using an algebraic method. You must show all steps of the algebraic process used. 5 + = = 4 Solve for in the first equation b subtracting 5 from both sides to get = Substitute this into the second equation to get 4 + 6( 5 39) = ( 5 39) = = 4 34 = 238 = 7 Simplif Add 234 to both sides Divide b 34 on both sides Thus = 5( 7) 39 = 4. Therefore the unique solution to the sstem of linear equation is = 7 and = Solve the following sstem of linear equations. Give a parameterization of the solutions if there are an infinite + 3 = 5 0 = 45 Add 5 times the first equation to the second equation to get 25 = 50. Dividing both sides of this equation b 25 gives = 2. Substituting = 2 into the first equation gives + 3(2) =, i.e., + 6 =. Solving this equation b subtracting 6 from both sides gives = 5. Therefore the unique solution is = 5 and = Solve the following sstem of linear equations. Give a parameterization of the solutions if there are an infinite = = 20 Notice that 5 3 times the first equation gives the second equation. This means that there are an infinite number of solutions. To find the parameterization solve for in either equation to get = So the solution set is parameterized b = t and = 2 3 t Solve the following sstem of linear equations. Give a parameterization of the solutions if there are an infinite = = 8 Notice that 6 7 times the first equation gives the second equation. This means that there are an infinite number of solutions. To find the parameterization solve for in either equation to get = 3 2. So the solution set is parameterized b = 3 2t and = t.

2 Review for Eam II Solutions MAT 2 5. A package deliver service charges a base price for deliver of packages weighing 5 pounds or less, and a surcharge for each additional pound. A customer is billed $3.75 for shipping a 7 pound package and $36.50 for shipping a 20 pound package. Find the base price and the surcharge for each additional pound. Let be the base price and be the surcharge for each additional pound. Then we need to solve the sstem of linear equations + 2 = = The base price is $0.25 and the surcharge is $ Animals in an eperiment are to be kept under a strict diet. Each animal should receive 5 grams of protein and 5 grams of fat. The laborator technician is able to purchase two food mies: Mi A has 25% protein and 0% fat; mi B has 36% protein and 2% fat. How man grams of each mi should be used to obtain the right diet for one animal? Let be the number of grams of mi A in the diet, and be the number of grams of mi B in the diet. Then we need to solve the sstem of linear equations = = 5 There need to be 00 grams of mi A and 250 grams of mi B in the diet. 7. Is the following matri in reduced form? Yes 8. Is the following matri in reduced form? No 9. Assume the following matri is the reduced form of a sstem of linear equations. How man solutions does the sstem have? There are no solutions 0. Assume the following matri is the reduced form of a sstem of linear equations. How man solutions does the sstem have? There are an infinite number of solutions Perform the row operation indicated b 5R + R 2 R 2 on the following matri. [ ] [ ] 3 4 5R + R 3 R Page 2

3 Review for Eam II Solutions MAT 2 2. Perform the row operation R R 2 on the following matri. [ ] 3 4 R R [ 6 0 ] Perform the row operation 6 R 2 R 2 on the following matri. [ ] R 2 R [ 3 ] Identif the row operations performed below. [ ] [ ] [ ] [ ] a b 4 2 c [ ] [ ] d 4 2 e a) R R 2 b) 8 R R c) 7R + R 2 R 2 d) 2 R 2 R 2 e) 4R 2 + R R 5. Graph the solution set to the sstem of linear inequalities and list the corner points of the solution (0, 4) (0, 0) (9,5) (0,0),(0,4),(9,5) (4, 0) (8, 0) 6. Graph the solution set to the sstem of linear inequalities and list the corner points of the solution (0, 5) (0,6) (8,3) (0, 0) (6, 0) (0,5),(8,3),(6,0) Page 3

4 Review for Eam II Solutions MAT 2 7. The graph of the sstem of linear inequalities is (0,3) 3 ( ) 0, (,2) (3,0) (5,0) Shade the feasible region and list the corner points of the solution set. The region has been shaded above.,(0, 5 2 ),(,2),(3,0). 8. Find the maimimum value of P = subject to the constraint (3, 8) The graph of the constraint is (,5) (6,) P = 39 at (,5), P = 3 at (6,), and P = 68 at (3,8). Thus the maimum value of P = 4+7 subject to the given constraint is P = 68 when = 3 and = Find the maimum value of P = 9 3 on the region graphed below. Also, give the and coordinates of the point in the region at which this maimum value occurs. (0, 200) (0, 75) (30, 50) P = 225 at (0,75), P = 20 at (30,50), P = 360 at (40,0), and P = 0 at. Thus the maimum value of P = 9 3 on the region graphed is P = 360 when = 40 and = 0. (40, 0) (90, 0) Page 4

5 Review for Eam II Solutions MAT Find the minimum value of P = on the region graphed below. Also, give the and coordinates of the point in the region at which this minimum value occurs. (0,6) ( ) 0, 32 (,4) 7 (3,0) (8,0) P = 56 at the verte (0,6), P = 36 at the verte (,4), and P = 256 at the verte (8,0). This is an unbounded region, and so we need to check if P = 36 reall is a minimum. At the point (,5), P = 62. At the point (2,4), P = 68. In both directions P increases, so P = 36 is a minimum. Thus the minimum value of P = on the region graphed is P = 36 when = and = A rancher raises emus and ostriches on his 60-acre farm. Each emu needs 5 acres of land and requires $500 of veterinar care, while each ostrich needs 6 acres of land and requires $300 of veterinar care. The rancher can afford no more than $4500 of veterinar care. If the epected profit is $50 for each emu and $70 for each ostrich, how man of each animal should he raise to obtain the greatest possible profit? Let be the number of emus and be the number of ostriches. Maimize P = subject to (0, 5) (0, 0) Graph of constraint: P = 700 at (0,0), P = 650 at (6,5), P = 450 at (0,9), and P = 0 at. Thus the maimum profit is $700 when the rancher raises (6,5) zero emus and ten ostriches. (9,0) (2, 0) 22. A farmer grows wheat and barle on her 80-acre farm. Each acre of wheat requires 3 da of labor per ear, and each acre of barle requires 2 das of labor per ear. The farmer can provide no more than 20 das of labor this ear. If the epected profit is $00 for each acre of wheat and $80 for each acre of barle, then how man acres of each should the farmer grow to obtain the greatest possible profit? Let be the number of acres of wheat, and be the number of acres of barle that the farmer can grow. Maimize P = subject to Graph of constraint: P = 7000 at (70,0), P = 7400 at (50,30), P = 2400 at (0,80), (0, 05) and P = 0 at, Thus the maimum profit is (0, 80) $7,400 when the farmer grows 50 acres of wheat and 30 acres (50, 30) of barle. (70, 0) (80, 0) Page 5

Intermediate Math Circles Wednesday November Inequalities and Linear Optimization

WWW.CEMC.UWATERLOO.CA The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Intermediate Math Circles Wednesda November 21 2012 Inequalities and Linear Optimization Review: Our goal is to solve sstems