6-1 Practice Multivariable Linear Systems and Row Operations

Transcription

1 6-1 Practice Multivariable Linear Systems and Row Operations Write each system of equations in triangular form using Gaussian elimination. Then solve the system [ ] 2. [ ] 3. [ ] Write the augmented matrix for each system of linear equations. 4. 5x 2y = x + 4y + 7z = x 2y z = 5 3x + y = 7 2x 3y + z = 6 2x z = 8 5x 2y + z = 4 y 2z = 4 Solve each system of equations using Gauss Jordan elimination. 7. 4x 2y = x 5y + z = x y + 3z = 38 x + 3y = 11 3x + 2y 4z = 1 2x + 5y 4z = 32 5x y + 2z = 6 x y + z = FRUIT Three customers bought fruit at Michael s Groceries. The table shows the amount of fruit bought by each person. Write and solve a system of equations to determine the price of each type of fruit. Name Apples Oranges Pears Total Cost ($) Rosario Lindsay Edwin Chapter 6 7 Glencoe Precalculus

2 6-2 Practice Matrix Multiplication, Inverses, and Determinants Find AB and BA, if possible. 1. A = [ ], B = [ ] 2. A = [ ], B = [ ] 3. GOLF The number of golf clubs manufactured daily by two different companies is shown, as well as the selling price of each type of club. Use this information to determine which company s daily production has the highest retail value. How much greater is the value? Company Club Type and Quantity 1-Wood 3-Wood 5-Wood Putter A B Club Club Value ($) 1-Wood Wood Wood 150 Putter 120 Write each system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve for X. 4. x 1 2x 2 + 3x 3 = x 1 + x 2 + 2x 3 = 11 5x 1 + 3x 2 x 3 = 13 5x 1 x 2 + 4x 3 = 1 4x 1 x 2 + 4x 3 = 11 3x 1 2x 2 + 8x 3 = 28 Determine whether A and B are inverse matrices. 6. A = [ ], B = [ 3 2 ] 7. A = [5 2 0 ], B = [ ] Find the determinant of each matrix. Then find its inverse, if it exists. 8. [ ] 9. [ ] Evaluate. A = [ ] B = [ 4 ] C = [ ] 10. AB + C 11. A(B C) Chapter 6 13 Glencoe Precalculus

3 6-3 Practice Solving Linear Systems Using Inverses and Cramer s Rule Use an inverse matrix to solve each system of equations, if possible. 1. 4x 7y = x 8y = 36 6x + 2y = 11 4x + 3y = 7 3. x 2y + 7z = x + y 2z = 5 4x + 5y z = 18 x + 2y + z = 8 5x 3y = 11 2x + 3y z = 1 5. TELEVISION During the summer, Manuel watches television M hours per day, Monday through Friday. Harry watches television H hours per day, Friday and Saturday. Ellen watches television E hours per day, Friday through Sunday. Altogether, they watch television 37 hours each week. On Fridays, they watch a total of 11 hours of television. If the number of hours Ellen spends watching television on any given day is twice the number of hours that Manuel spends watching television on any given day, how many hours of television does each of them watch each day? Use Cramer s Rule to find the solution of each system of linear equations, if a unique solution exists. 6. 4x 5y = 1 7. x + y + z = 8 2x 3y = 1 3x z = 22 y + 2z = PAPER ROUTE Payton, Santiago, and Queisha each have a paper route. Payton delivers 5 times as many papers as Santiago. Santiago delivers twice as many papers as Queisha. If 20 papers were added to Payton s route, he would then deliver four times the total number of papers that Santiago and Queisha deliver. How many papers does each person deliver? Chapter 6 18 Glencoe Precalculus

4 6-4 Practice Partial Fractions Find the partial fraction decomposition of each rational expression. 1. 3x 7 x 2 7x x2 10x 2 x 3 + x 2 2x 3. 9x + 15 x 2 + 3x x 2x 2 9x + 9 Find the partial fraction decomposition of each improper rational expression. 5. 3x2 + 5x + 2 x 2 + 2x 6. 5x2 11x + 54 x 2 + 2x x2 + 17x + 2 x 2 + x 8. 8x2 + 22x 10 (2x 3) 2 Find the partial fraction decomposition of each rational expression with repeated factors. 9. 2x2 + 29x 100 x 3 10x x 10. 5x4 7x 3 12x 2 + 6x + 21 (x 3)(x 2 2) x x 3 + 6x 2 + 9x 12. 4x4 + 8x 3 + 6x 2 + 6x + 5 (3x + 2)(x 2 + 1) GROWTH When working with exponential growth in calculus, it is often necessary to work with functions of the 1 form f(x) = and to decompose these functions into the sum of its partial fractions. Find the partial x(50 x) decomposition of f(x). Chapter 6 24 Glencoe Precalculus

5 6-5 Practice Linear Optimization Find the maximum and minimum values of the objective function f(x, y) and for what values of x and y they occur, subject to the given constraints. 1. f(x, y) = 2x + 5y 2. f(x, y) = 4x + 3y x 0 x 0 y 0 y 0 x + y 7 2x + 3y 6 2x + 3y 18 x + y 8 3. f(x, y) = 2x 3y 4. f(x, y) = 3x + 3y x 0 x 0 x 7 y 0 y 0 y 8 y 5 x + y 10 x + 2y 14 3x + 2y SKATES A manufacturer produces roller skates and ice skates. Manufacturer Information Roller Skates Ice Skates Maximum Time Available Assembling 5 minutes 4 minutes 200 minutes Checking and Packaging 1 minute 4 minutes 120 minutes Profit per Skate $40 $30 a. Write an objective function and list the constraints that model the given situation. b. Sketch a graph of the region determined by the constraints from part a to find the set of feasible solutions for the objective function. c. How many roller skates and ice skates should be manufactured to maximize profit? What is the maximum profit? d. Describe why the company would choose a number of roller skates and ice skates different from the answer in part c. Chapter 6 29 Glencoe Precalculus