Practice Test - Chapter 6
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1 1. Write each system of equations in triangular form using Gaussian elimination. Then solve the system. Align the variables on the left side of the equal sign. Eliminate the x-term from the 2nd equation. Multiply the 1st equation by 5 and the 2nd equation by 3. Add the equations together. Replace the 2nd equation with y = 5. The system is now in triangular form. Substitute for y and solve for x. The solution is. 2. Align the variables on the left side of the equal sign. Eliminate the x-term from the 2nd equation. Multiply the 1st equation by 5. Add the equations together. esolutions Manual - Powered by Cognero Page 1
2 The new set of equations is: Eliminate the x-term from the 3rd equation. Multiply the 1st equation by 3. Add the equations together. The new set of equations is: We can simplify the 2nd equation by dividing by 9. Eliminate the y-term from the 3rd equation. Multiply the 2nd equation by 14 and the third equation by 3. Add the equations together. Replace the 3rd equation with z = 8. The system is now in triangular form. Substitute for z and solve for y, then substitute for y and z and solve for x. esolutions Manual - Powered by Cognero Page 2
3 3. The solution is. Solve the system of equations. Write the augmented matrix. Apply elementary row operations to obtain reduced row-echelon form. The solution is (2, 3). esolutions Manual - Powered by Cognero Page 3
4 4. Write the augmented matrix. Apply elementary row operations to obtain reduced row-echelon form. The solution is. esolutions Manual - Powered by Cognero Page 4
5 5. LIBRARY Kristen checked out books, CDs, and DVDs from the library. She checked out a total of 16 items. The number of CDs and DVDs equaled the number of books. She checked out two more CDs than DVDs. a. Let b = number of books, c = number of CDs, and d = number of DVDs. Write a system of three linear equations to represent the problem. b. Solve the system of equations. Interpret your solution. a. Use the total number of items for the first equation: b + c + d = 16. The total number of books equaled the sum of the number of CDs and DVDs: b = c + d. She checked out 2 more CDs than DVDs: c = d + 2. Rewrite the equations, aligning the variables. b. Write the augmented matrix. Apply elementary row operations to obtain reduced row-echelon form. Kristen checked out 8 books, 5 CDs, and 3 DVDs. esolutions Manual - Powered by Cognero Page 5
6 Find AB and BA, if possible. 6. A =, B = A = ; B = A is a 4 4 matrix and B is a 4 2 matrix. Because the number of columns of A is equal to the number of rows of B, AB exists. To find the first entry of AB, find the sum of the products of the entries in row 1 of A and column 1 of B. Follow the same procedure for row 2 column 1 of AB and the remaining entries. Because the number of columns of B is not equal to the number of rows of A, BA is undefined. esolutions Manual - Powered by Cognero Page 6
7 7. A =, B = A = ; B = A is a 3 3 matrix and B is a 1 3 matrix. Because the number of columns of A is not equal to the number of rows of B, AB is undefined. B is a 1 3 matrix and A is a 3 3 matrix. Because the number of columns of B is equal to the number of rows of A, BA exists. To find the first entry of BA, find the sum of the products of the entries in row 1 of B and column 1 of A. Follow the same procedure for row 2 column 1 of BA and the remaining entries. 8. GEOMETRY The coordinates of a point (x, y) can be written as a 2 1 matrix. Let A = a. Let P be the point ( 3, 4). Discuss what effect multiplying A by P has on P. b. A triangle contains vertices (0, 0), (2, 6), and (8, 3). Create B, a 2 3 matrix to represent the triangle. Find AB. What is the effect on the triangle? Does it agree with your answer to part a? a. P =. P = ( 3, 4) and AP = ( 4, 3). Plot these points on a graph. It appears that AP is P rotated 90 degrees counterclockwise. This angle can be confirmed in a variety of ways. esolutions Manual - Powered by Cognero Page 7
8 The distance from P to the origin is 5. The distance from AP to the origin is also 5. Find the distance from P to AP. The sides 5, 5, and 5 form a right triangle because. Therefore, multiplying P by A rotates the point 90 counterclockwise. b. Let each point be a column. B = Find AB. Draw both triangles. It appears that AB is B rotated 90 degrees counterclockwise. This angle can be confirmed by checking the rotation of one side of the triangle. In this case, we will check the rotation of side h (side g in the rotation). Sides g and h form a right triangle. Find the length of each side. These sides form a right triangle, so triangle A is rotated 90 counterclockwise. This agrees with answer a. Find A 1, if it exists. If A 1 does not exist, write singular. esolutions Manual - Powered by Cognero Page 8
9 9. Create the doubly augmented matrix. Apply elementary row operations to write the matrix in reduced row-echelon form. The first two columns are the identity matrix. Therefore, A is invertible and A 1 =. Confirm that AA 1 = A 1 A = I. esolutions Manual - Powered by Cognero Page 9
10 10. Create the doubly augmented matrix. Apply elementary row operations to write the matrix in reduced row-echelon form. esolutions Manual - Powered by Cognero Page 10
11 The first two columns are the identity matrix. Therefore, A is invertible and A 1 =. Confirm that AA 1 = A 1 A = I. esolutions Manual - Powered by Cognero Page 11
12 Confirm that AA = A A = I. esolutions Manual - Powered by Cognero Page 12
13 11. Use an inverse matrix to solve each system of equations, if possible. Write the system in matrix form AX = B. Use the formula for the inverse of a 2 2 matrix to find A 1. Multiply A 1 by B to solve the system. The solution is (1, 3) esolutions Manual - Powered by Cognero Page 13
14 12. Write the system in matrix form AX = B. Use a graphing calculator to find A 1. Enter the values for the matrix A. Enter [A] 1. Select values in reduced fraction form. to get the Multiply A 1 by B to solve the system. esolutions Manual - Powered by Cognero Page 14
15 Multiply A by B to solve the system. The solution is ( 9, 1, 2). esolutions Manual - Powered by Cognero Page 15
16 Use Cramer s Rule to find the solution of each system of linear equations, if a unique solution exists. 13. The coefficient matrix is A =. Calculate the determinant of A. det(a) = A = = ad bc Because the determinant of A is not 0, you can apply Cramer s Rule. Remember to replace the column associated with the variable with the column of constant terms. The solution is (4, 7). 14. esolutions Manual - Powered by Cognero Page 16
17 The coefficient matrix is A =. Remember to align the terms, using 0s if needed. Calculate the determinant of A. det(a) = A = = Because the determinant of A is not 0, you can apply Cramer s Rule. Remember to replace the column associated with the variable with the column of constant terms. esolutions Manual - Powered by Cognero Page 17
18 The solution is ( 4, 3, 2). esolutions Manual - Powered by Cognero Page 18
19 15. Find the partial fraction decomposition of each rational expression. Rewrite the expression as partial fractions with constant numerators, A and B, and denominators that are the linear factors of the original denominator. Multiply each side by the LCD, x 2 9. Group the like terms. Equate the coefficients on the left and right side of the equation to form two equations with 2 variables. In other words, the coefficients of the x-terms on the left side of the equation must equal the coefficients of the x-terms on the right side. Use any method to solve the new system. Replace A and B with 2 and 2 in the partial fraction decomposition. esolutions Manual - Powered by Cognero Page 19
20 16. Rewrite the expression as partial fractions with constant numerators, A and B, and denominators that are the linear factors of the original denominator. Multiply each side by the LCD, x 2 4x + 3. Group the like terms. Equate the coefficients on the left and right side of the equation to form two equations with 2 variables. In other words, the coefficients of the x-terms on the left side of the equation must equal the coefficients of the x-terms on the right side. Use any method to solve the new system. Replace A and B with 8 and 6 in the partial fraction decomposition. esolutions Manual - Powered by Cognero Page 20
21 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. 17. Begin by graphing the given system of inequalities. The solution of the system, which makes up the set of feasible solutions for the objective function, is the shaded region, including its boundary segments. The polygonal region of feasible solutions has three vertices. Each vertex is located at an intersection of two of the boundaries. The vertices are (0, 0), (4, 0), and (0, 8). Find the value of the objective function for each of the vertices. max at (4, 0) = 8, min at (0, 8) = Begin by graphing the given system of inequalities. The solution of the system, which makes up the set of feasible solutions for the objective function, is the shaded region, including its boundary segments. Rewrite the inequalities in slope-intercept form. esolutions Manual - Powered by Cognero Page 21
22 The polygonal region of feasible solutions has three vertices. Each vertex is located at an intersection of two of the boundaries. Two of the vertices are (0, 0) and (0, 9). Find the third vertex by solving the system of equations that form the intersection. The third vertex is (27, 9). Find the value of the objective function for each of the vertices. max at (0, 9) = 18, min at (27, 9) = 9 esolutions Manual - Powered by Cognero Page 22
23 19. PRICING The Harvest Nut Company sells create-your-own trail mixes where customers can choose whatever combinations they want. Colin s favorite mix contains peanuts, dried cranberries, and carob-coated pretzels. The prices for each are shown below. If Colin bought a 5-pound mixture for $16.80 that contained twice as many pounds of carob-coated pretzels as cranberries, how many pounds of each item did he buy? Let p = peanuts, r = raisins, and z = pretzels. Set up a system of equations. He bought a five-pound mixture, so p + r + z = 5. He spent $16.80, so 3.2p + 2.4r + 4z = There were twice as many pretzels as raisins, so z = 2r. Write the system of equations, aligning the variables. Write the augmented matrix. Apply elementary row operations to obtain reduced row-echelon form. peanuts, 2 lb; cranberries, 1 lb; carob-coated pretzels, 2 lb esolutions Manual - Powered by Cognero Page 23
24 20. MULTIPLE CHOICE The graph displays the constraints for an objective function. Which of the following CANNOT be one of the constraints? A y 0 B x 0 C x y 0 D x y 0 Identify which inequalities are constraints. Choice A, y 0, represents the bottom line of the shaded region. Choice B, x 0, can be a constraint even though it has no direct effect on the boundaries of the shaded region. Choice D, x y 0, represents the diagonal boundary of the shaded region. Choice C, x y 0, cannot be a constraint because none of the possible values from the constraint fall in the region. The correct choice is C. esolutions Manual - Powered by Cognero Page 24
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