Math 370 Exam 4 Review Name SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the system of equations by the substitution method. 1) y = 4x - 5 y = 7x - 6 1 3, - 11 3 Objective: (7.1) Solve Linear Systems by Substitution 2) 2x - 6y = 10-2x 6x + 4y = x + 3y + -13 {(-2, -3)} Objective: (7.1) Solve Linear Systems by Substitution 1) 2) Solve the system by the addition method. 3) x + y = -2 x - y = 15 {(6.5, -8.5)} Objective: (7.1) Solve Linear Systems by Addition 4) 9x + 6y = 51 3x - 2y = 25 {(7, -2)} Objective: (7.1) Solve Linear Systems by Addition 3) 4) Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 5) x + y = -6 5) x + y = -7 Objective: (7.1) Identify Systems That Do Not Have Exactly One Ordered-Pair Solution 6) 5x + y = 7 3y = 21-15x {(x, y) 5x + y = 7} Objective: (7.1) Identify Systems That Do Not Have Exactly One Ordered-Pair Solution 6) Solve the problem. 7) One number is 5 less than a second number. Twice the second number is 8 less than 4 times the first. Find the two numbers. 9 and 14 8) A chemist needs 190 milliliters of a 64% solution but has only 52% and 71% solutions available. How many milliliters of each should be mixed to obtain the desired solution? 70 ml of 52%; 120 ml of 71% 7) 8)
9) The manager of a bulk foods establishment sells a trail mix for $7 per pound and premium cashews for $12 per pound. The manager wishes to make 100 pounds of a trail mix-cashew mixture that will sell for $11 per pound. How many pounds of each should be used? 20 pounds of trail mix 80 pounds of cashews 10) A plane flies 490 miles with the wind and 310 miles against the wind in the same length of time. If the average velocity of the wind is 27 mph, what is the average velocity of the plane in still air? 120 mph 11) You invested $21,000 and started a business selling vases. Supplies cost $15 per vase and you are selling each vase for $30. Determine the number of vases, x, that must be produced and sold to break even. 1400 units 9) 10) 11) Solve the system of equations. 12) x + y + z = 3 x - y + 2z = 9 5x + y + z = 15 {(3, -2, 2)} Objective: (7.2) Solve Systems of Linear Equations in Three Variables 13) 3x + 2y + z = 9 2x - 4y - z = -5 3x + y + 2z = 3 {(2, 3, -3)} Objective: (7.2) Solve Systems of Linear Equations in Three Variables 12) 13) Solve the problem. 14) The following is known about three numbers: If the second number is subtracted from the sum of the first number and 4 times the third number, the result is -19. The third number plus 2 times the first number is -5. The first number plus 2 times the second number plus the third number is -7. Find the three numbers. [Hint: let x represent the first number, y the second number, and z the third number. Use the given conditions to write and solve a system of equations.] x = 0, y = -1, z = -5 Objective: (7.2) Solve Problems Using Systems in Three Variables 15) Three shrimp boats supply the shrimp wholesalers on Hilton Head with fresh catch. The Annabelle takes 50% of its catch to Hudson's, 20% to Captain J's, and 30% to Mainstreet. The Curly Q takes 40% of its catch to Hudson's, 40% to Captain J's, and 20% to Mainstreet. The SloJoe takes 30% of its catch to Hudson's, 40% to Captain J's, and 30% to Mainstreet. One week Hudson's received 268.2 pounds of shrimp, Captain J's received 235.6 pounds, and Mainstreet received 173.2 pounds. How many pounds of shrimp did each boat catch? Annabelle 176 lbs, Curly Q 299 lbs, SloJoe 202 lbs Objective: (7.2) Solve Problems Using Systems in Three Variables 14) 15)
Write the partial fraction decomposition of the rational expression. 16) 5x 2 - x - 18 x3 - x 16) 18 x + -6 x + 1 + -7 x - 1 Objective: (7.3) Decompose P/Q, Where Q Has Only Distinct Linear Factors 17) x + 2 x3-2x2 + x 2 x + -2 x - 1 + 3 (x - 1)2 Objective: (7.3) Decompose P/Q, Where Q Has Repeated Linear Factors 17) 18) 6x 2-10x + 26 (x - 2)(x2 + 6) 18) 3 x - 2 + 3x - 4 x2 + 6 Objective: (7.3) Decompose P/Q, Where Q Has a Nonrepeated Prime Quadratic Factor 19) 2x 3 + 3x2 (x2 + 5) 2 19) 2x + 3 x2 + 5 + -10x - 15 (x2 + 5) 2 Objective: (7.3) Decompose P/Q, Where Q Has a Prime, Repeated Quadratic Factor Solve the system by the substitution method. 20) x2 + y2 = 113 x + y = 15 {(8, 7), (7, 8)} Objective: (7.4) Solve Nonlinear Systems By Substitution 20) Solve the system by the addition method. 21) 8x2 + y2 = 64 8x2 - y2 = 64 {(2 2, 0), (-2 2, 0)} Objective: (7.4) Solve Nonlinear Systems By Addition 21) 22) x2-3y2-1 = 0 4x2 + 3y2-19 = 0 {(2, 1), (2, -1), (-2, 1), (-2, -1)} Objective: (7.4) Solve Nonlinear Systems By Addition 22)
Solve the problem. 23) A right triangle has an area of 30 square inches. The square of the hypotenuse is 136. Find the lengths of the legs of the triangle. Round your answer to the nearest inch. 6 inches and 10 inches Objective: (7.4) Solve Problems Using Systems of Nonlinear Equations 23) Graph the solution set of the system of inequalities or indicate that the system has no solution. 24) 2x - y -4 x + 4y 4 24) Objective: (7.5) Graph a System of Inequalities
25) x2 + y2 36-8x + 3y -24 25) Objective: (7.5) Graph a System of Inequalities Solve the system of equations using matrices. Use Gaussian elimination with back-substitution. 26) x + y + z = -5 x - y + 3z = -1 4x + y + z = -2 {(1, -4, -2)} Objective: (8.1) Use Matrices and Gaussian Elimination to Solve Systems 26) Solve the system of equations using matrices. Use Gauss-Jordan elimination. 27) x = -1 - y - z x - y + 2z = -1 4x + y =14 - z {(5, -2, -4)} Objective: (8.1) Use Matrices and Gauss-Jordan Elimination to Solve Systems 27) Evaluate the determinant. 28) 9-6 8 1 57 Objective: (8.5) Evaluate a Second-Order Determinant 28)
Use Cramer's rule to solve the system. 29) 5x = -5y + 10 2x = -y - 2 {(-4, 6)} Objective: (8.5) Solve a System of Linear Equations in Two Variables Using Cramer's Rule 29) Evaluate the determinant. 30) 2 4 5 2 6 4 2 5 3-6 Objective: (8.5) Evaluate a Third-Order Determinant 30) Solve the problem. 31) Determinants are used to show that three points lie on the same line (are collinear). If x1 y1 1 x2 y2 1 x3 y3 1 = 0, then the points (x1, y1), (x2, y2), and (x3, y3) are collinear. If the determinant does not equal 0, then the points are not collinear. Are the points (-2, -1), (0, 9), and (-6, -21) collinear? Yes Objective: (8.5) Evaluate a Third-Order Determinant 31) Use Cramer's rule to solve the system. 32) 9x - 5y - z = 22 x - 2y + 5z = 15-9x + y + z = -58 {(8, 9, 5)} Objective: (8.5) Solve a System of Linear Equations in Three Variables Using Cramer's Rule 32)