Study Guide and Review - Chapter 6
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1 State whether each sentence is or false. If false, replace the underlined term to make a sentence. 1. If a system has at least one solution, it is said to be consistent. Graph each system and determine the number of solutions that it has. If it has one solution, name it. 9. x y = 1 x + y = 5 one; (3, 2) 2. If a consistent system has exactly two solution(s), it is said to be independent. false; one 3. If a consistent system has an infinite number of solutions, it is said to be inconsistent. false; dependent 4. If a system has no solution, it is said to be inconsistent. 10. y = 2x 4 4x + y = 2 one; (1, 2) 5. Substitution involves substituting an expression from one equation for a variable in the other. 6. In some cases, dividing two equations in a system together will eliminate one of the variables. This process is called elimination. false; adding or subtracting 7. A set of two or more inequalities with the same variables is called a system of equations. false; system of inequalities 8. When the graphs of the inequalities in a system of inequalities do not intersect, there are no solutions to the system. esolutions Manual - Powered by Cognero Page 1
2 11. 2x 3y = 6 y = 3x + 2 one; (0, 2) 14. 3x + y = 5 6x = 10 2y infinitely many solutions 12. 3x + y = 3 y = x 3 one; (0, 3) 15. MAGIC NUMBERS Sean is trying to find two numbers with a sum of 14 and a difference of 4. Define two variables, write a system of equations, and solve by graphing. Sample answer: Let x be one number and y the other number; x + y = 14; x y = 4; 9 and x + 2y = 6 3x + 6y = 8 no solution Use substitution to solve each system of equations. 16. x + y = 3 x = 2y (2, 1) 17. x + 3y = 28 y = 5x (2, 10) esolutions Manual - Powered by Cognero Page 2
3 18. 3x + 2y = 16 x = 3y 2 (4, 2) 19. x y = 8 y = 3x (2, 6) 20. y = 5x 3 x + 2y = 27 (3, 12) 21. x + 3y = 9 x + y = 1 ( 3, 4) 22. GEOMETRY The perimeter of a rectangle is 48 inches. The length is 6 inches greater than the width. Define the variables, and write equations to represent this situation. Solve the system by using substitution. Sample answer: Let w be the width and l be the length; 2l + 2w = 48, l = w + 6; 9 is the width and 15 is the length. Use elimination to solve each system of equations. 23. x + y = 13 x y = 5 (9, 4) 24. 3x + 4y = 21 3x + 3y = x + 4y = 4 x + 10y = 16 (4, 2) 26. 2x + y = 5 x y = 2 ( 1, 3) 27. 6x + y = 9 6x + 3y = x 4y = 2 3x + 4y = 38 (10, 2) 29. 2x + 2y = 4 2x 8y = 46 ( 3, 5) 30. 3x + 2y = 8 x + 2y = BASEBALL CARDS Cristiano bought 24 baseball cards for $50. One type cost $1 per card, and the other cost $3 per card. Define the variables, and write equations to find the number of each type of card he bought. Solve by using elimination. Sample answer: Let f be the number of the first type of card, and let c be the number of the second type of card; f + c = 24, f + 3c = 50; 11 $1 cards and 13 $3 cards. esolutions Manual - Powered by Cognero Page 3
4 Use elimination to solve each system of equations. 32. x + y = 4 2x + 3y = BAKE SALE On the first day, a total of 40 items were sold for $356. Define the variables, and write a system of equations to find the number of cakes and pies sold. Solve by using elimination. (1, 3) 33. x y = 2 2x + 4y = 38 (5, 7) 34. 3x + 4y = 1 5x + 2y = 11 (3, 2) 35. 9x + 3y = 3 3x 2y = 4 (2, 5) 36. 8x 3y = 35 3x + 4y = 33 ( 1, 9) 37. 2x + 9y = 3 5x + 4y = 26 (6, 1) 38. 7x + 3y = 12 2x 8y = 32 (0, 4) 39. 8x 5y = 18 6x + 6y = 6 (1, 2) Sample answer: Let c represent the number of the cakes, and let p represent the number of pies; 8c + 10p = 356, p + c = 40; 22 cakes, 18 pies Determine the best method to solve each system of equations. Then solve the system. 41. y = x 8 y = 3x Subs.; (2, 6) 42. y = x y = 2x Subs.; (0, 0) 43. x + 3y = 12 x = 6y Subs.; (24, 4) 44. x + y = 10 x y = 18 Elim (+); (14, 4) esolutions Manual - Powered by Cognero Page 4
5 45. 3x + 2y = 4 5x + 2y = 8 Elim ( ); ( 2, 1) 46. 6x + 5y = 9 2x + 4y = 14 Elim ( ); ( 1, 3) 47. 3x + 4y = 26 2x + 3y = 19 Elim ( ); (2, 5) x 6y = 3 5x 8y = 25 Elim ( ); (3, 5) 49. COINS Tionna has 25 coins in her piggy bank with a value of $4. The coins are either dimes or quarters. Define the variables, and write a system of equations to determine the number of dimes and quarters. Then solve the system using the best method for the situation. Sample answer: Let d represent the number of dimes and let q represent the number of quarters; d + q = 25, 0.10d q = 4; 15 dimes, 10 quarters 50. FAIR At a county fair, the cost for 4 slices of pizza and 2 orders of French fries is $ The cost of 2 slices of pizza and 3 orders of French fries is $ To find out how much a single slice of pizza and an order of French fries costs, define the variables and write a system of equations to represent the situation. Determine the best method to solve the system of equations. Then solve the system. Let p represent the cost of a slice of pizza and t represent the cost of an order of French fries; 4p + 2f = 21, 2p + 3f = 16.5; Sample answer: elimination; pizza $3.75; French fries $3. Solve each system of inequalities by graphing. 51. x > 3 y < x y 5 y > x 4 esolutions Manual - Powered by Cognero Page 5
6 53. y < 3x 1 y 2x y x 3 y 3x JOBS Kishi makes $7 an hour working at the grocery store and $10 an hour delivering newspapers. She cannot work more than 20 hours per week. Graph two inequalities that Kishi can use to determine how many hours she needs to work at each job if she wants to earn at least $90 per week. esolutions Manual - Powered by Cognero Page 6
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