Name: ______________________
Class: _________________
Date: _________
ID: A
Algebra 1: Spring Final Exam Practice Test This practice test is broken into two large section and then each of those has sample problems of each type. - Multiple Choice: -Short Answer: Topics covered: ~ Solve a system of equations by graphing ~ Solve systems of equations using substitution ~ Solve systems of equations using elimination ~ Solve systems of equations through either method you wish ~ Distribute expressions using FOIL or other appropriate methods (we went over three) ~ Factor quadratics ~ Solve quadratics
____________________________________________________________________________ Multiple Choice Identify the choice that best completes the statement or answers the question. Solve the system using Graphing 1. The Stevens family is going to the county fair. They have two ticket options as shown in the chart below. T icket Option A
Admission Price $6
Price Per Ride $ .30
B $4 $ .70 I. Write an equation that shows the cost per person for each option. II. Use graphing to solve the system of equations. III. Find the number of rides for which the total cost is the same with both ticket options. a. I. C = 6 + 0.3r c. I. C = 6 + 0.3r C = 4 + 0.7r C = 4 + 70r II. (5, 7.5) II. (5, 7.5) III. 5 rides III. 5 rides b. I. C = 4 + 0.7r d. I. C = 6 + 30r C = 6 + 30r C = 4 + 70r II. (0.05, 0.075) II. (0.05, 0.075) III. 0.05 ride III. 0.005 ride
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Name: ______________________
ID: A
2. Lena made 32 ounces of a fruit drink mix using pineapple juice and grapefruit juice. The number of ounces of pineapple juice in the fruit drink mix is 5 more than 2 times the number of ounces of grapefruit juice in the fruit drink mix. Which graph shows the number of ounces pineapple juice, x, and the number of ounces of grapefruit juice, y, in the fruit drink mix? What system of equations was used to create the graph? a.
x + y = 32
c.
x− 5 = 2y
b.
5x + 2y = 32 x − 5 = 2y
x + y = 32
d.
x+ 5 = 2y
5x + 2y = 32 x + 5 = 2y
Solve the system using Substitution 3. Solve by substitution: 3x + 2y = − 4 y = 4x − 2 a. ÊÁË 0, − 2 ˆ˜¯ b. no solution c. ÊÁË 2, 6 ˆ˜¯
1 d. (–1, − ) 2
Solve using Elimination 4.
3x + 6y = 9 x − 6y = 11 a. (5, –1) b. (0,
3 1 ) c. (10, − ) d. no solution 2 6
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Name: ______________________
ID: A
Solve the system using either Substitution OR Elimination. 5. The length of a rectangle is 8 cm more than four times the width. If the perimeter of the rectangle is 46 cm, what are the dimensions? a. width = 3 cm, length = 20 cm b. width = 3 cm, length = 40 cm c. width = 6 cm, length = 32 cm d. width = 6 cm, length = 40 cm 6.
3x + y = 5 y = 5x − 5 a. ÊÁË 0, 5 ˆ˜¯ b. ÊÁË 1.25, 1.25 ˆ˜¯ c. ÊÁË 1.5, 0.5 ˆ˜¯ d. ÊÁË −3, 14 ˆ˜¯
7. Marc sold 461 tickets for the school play. Student tickets cost $3 and adult tickets cost $4. Marc's sales totaled $1624. How many adult tickets and how many student tickets did Marc sell? a. 220 adult, 241 student b. 225 adult, 236 student c. 236 adult, 225 student d. 241 adult, 220 student 6x + 4y =10
8.
18x + 12y =− 20 a. no solution b. (–7, 13) c. (7¸ –8) d. (–1, 4) 9. Which system of equations has no solution? a.
7x + 9y = 5 −21x − 27y =14
b.
7x + 9y = 5
c.
4x + 7y =14
7x − 9y = 10 7x − 36y =40
d.
7x − 9y = 5 14x − 19y = 10
10. Taylor has a pocket full of change with a combination of quarters and nickles. Taylor sees that there is a total of $4.50, but there is only 34 coins all together. How many of each kind of coin does Taylor have? a. x=14, y=20 b. x=20, y=14 11. Anita wants to buy some dvd’s and has $181.00. But since Anita needs to buy these for late Valentines day presents, Anita must get 13 dvd’s to please everyone. The discount rack of dvd’s cost $5.00 but the new releases cost $19.50. How many of each can Anita buy for Valentines? a. x=5, y=8 b. x=8, y=5 12. Your teacher is giving a test worth 200 points. There is a total of 30 five-point and ten-point questions. How many five-point questions are on the test? a. 10 b. 15 c. 20 d. 25 13. If x + 2y = 8 and −2x + 3y = 5, then x = ? a. 2 b.
13 5
c. 3 d. 5
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Name: ______________________
ID: A
Other types of “systems” problems 14. Which choice best describes the solution(s) of the system of equations? −24x + 8y = 24 −15x + 5y = 15 a. many solutions b. (–1, 0) is the only solution. c. (1, 48) is the only solution. d. no solution 15. Mr. Frankel bought 7 tickets to a puppet show and spent $43. He bought a combination of child tickets for $4 each and adult tickets for $9 each. Which system of equations below will determine the number of adult tickets, a, and the number of child tickets, c, he bought? a. a = c − 9 c. a + c = 301 9a + 4c = 43 a +c=7 b. 9a + 4c = 43 d. 4a + 4c = 50 a +c=7 a +c=7 Distribute (or FOIL when appropriate) 16. ( x + 5 ) ( x + 2 ) 2
2
2
2
a. x + 7x + 10 b. x − 7x − 10 c. x − 7x + 10 d. x + 7x − 10 Ê 2 ˆ 17. ( x + 7 ) ÁÁÁÁ x − 4x + 2 ˜˜˜˜ Ë ¯ 3
2
3
2
3
2
3
2
a. x + 3x − 26x + 14 b. x + 11x − 26x + 14 c. x + 3x − 30x + 14 d. x + 11x − 30x + 14 18. A rectangle has a length of x + 5 and a width of x − 7. Write an equation that describes the area, A, of the rectangle in terms of x. 2 2 a. A = x − 2x − 35 b. A = x + 12x − 35 c. A = 2x − 2 d. A = 2x + 12 Factor the quadratic 2
19. x + 6x + 5 a. ( x+ 1 ) ( x− 5 ) b. ( x− 1 ) ( x − 5 ) c. ( x+ 1 ) ( x+ 5 ) d. ( x − 1 ) ( x+ 5 ) 2
20. Factor: x + 4x − 77. a. (x + 4)(x − 77) b. (x − 7)(x + 11) c. (x + 7)(x − 11) d. (x − 7)(x − 11)
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Name: ______________________
ID: A
Short Answer Solve the system using Graphing 21. Find the solution to the system by graphing. x+y = 0 2x − y = − 9
22. Solve the system by graphing: x+y= 8 3x − y = 4
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Name: ______________________
ID: A
23. Solve the linear system by graphing. 2 y = x+2 3 y = −x − 3
Solve the system using either Substitution OR Elimination. 24. A rental car agency charges $15 per day plus 11 cents per mile to rent a certain car. Another agency charges $18 per day plus 8 cents per mile to rent the same car. How many miles will have to be driven for the cost of a car from the first agency to equal the cost of a car from the second agency? Express the problems as a system of linear equations and solve using the method of your choice. 25. The length of a rectangle is 7 cm more than four times the width. If the perimeter of the rectangle is 44 cm, what are its dimensions? 26. The sum of the ages of Petra and her mother is 53. Her mother is 11 years more than twice as old as Petra. How old are Petra and her mother? 27. At a high school basketball game, 400 tickets were sold. Adult tickets cost $5 and student tickets cost $2.50. If the total amount collected was $1375, how many student tickets were sold? 28.
x + 4y = −23 −3x + y = 4
29. The table below shows the costs of two different combinations of hot dogs and sodas at a ballgame. What is the cost h of one hot dog and the cost s of one soda? Number of hot dogs 4 4
Number of sodas 4 6
Total Cost $20 $24
30. Solve the linear system by any method. 5x − 2y = 3 −x + 6y = −2 31. A music store is selling compact discs for $11.50 and $7.50. You buy 12 discs and spend a total of $106. How many compact discs that cost $11.50 did you buy? 32. Reggie receives three points for each correct answer on a test and loses one point for each incorrect answer. a. Write an equation showing that Reggie received 60 points. b. Write an equation showing that Reggie answered all 40 questions on the test. c. Solve the system of equations you wrote for parts a and b by graphing. 6 /8
Name: ______________________
ID: A
33. A boat travels with the current at a speed of 10 miles per hour with respect to land, then against the same current at a speed of 6 miles per hour with respect to land. Find (a) the speed of the current, and (b) the speed of the boat in still water. Solve the system using Substitution 34. Use substitution to solve the linear system. x − 4y = 6 2x + y = −4 Solve using Elimination 35.
3x − 4y = 5 5x + 4y = −13
36. Use elimination to solve the linear system. 3x+2y= − 5 4x − 3y=16 Other types of “systems” problems 37. A mistake has been made in the solution. Explain the error and how to correct it. y = 2x − 2 5x − 2y = 18 5x − 2 ( 2x − 2 ) = 18 5x − 4x − 4 = 18 x − 4 = 18 x = 22 x = 22 y = 2 ( 22 ) − 2 y = 44 − 2 y = 42 Solution: x = 22 and y = 42 38. (2, 3)
−2x + 3y = 5 3x + 2y = 12
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Name: ______________________
ID: A
39. Express each equation in slope-intercept form. Then determine, without solving the system, whether the system of equations has exactly one solution, no solution, or an infinite number of solutions. 15x + 5y = 5 −6x − 2y = −2 Distribute (or FOIL when appropriate) 40. 20x(3 – 2x) 41. ( 5x + 7 ) ( 3x − 4 ) Factor the quadratic 2
42. x − 16x + 63 2
43. x − 7x + 12 2
44. 25x − 15x + 2 2
45. 5x + 43x+ 24 2
46. 4x − 4x− 3 Solve the quadratic equation 47. ( x + 8 ) ( x − 1 ) = 0 48. Solve the equation ( x− 9 ) ( x− 7 ) = 0. 2
49. x − 6x − 16 = 0 2
50. x + 6x − 27 = 0 2
51. Solve the equation 30x + 11x − 30 = 0. 2
52. Solve the equation 4x + 7x − 2 = 0. 53. A rectangle with an area of 24 square units has length x + 1 and width 4x − 6. Find the value of x. 54. The area of a rectangle is 24 square centimeters and its side lengths are x centimeters and x + 2 centimeters. a. Find the side lengths of the rectangle. Justify your answer. b. Find the perimeter of the rectangle. 55. Consider the equation ( 3t − 15 ) ( 4t + 22 ) = 0. a. Solve the equation. b. Susan noticed that she could factor out a 3 from the first expression on the left side of the equation and a 2 from the second expression. She rewrote the equation as 3 ( t − 5 ) ⋅ 2 ( t + 11 ) = 0. Is this equation equivalent to the original equation? Explain.
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ID: A
Algebra 1: Spring Final Exam Practice Test Answer Section MULTIPLE CHOICE 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
A A A A A B D A A A quarters=14, nickles=20 x + y = 34
25x + 5y = 450 11. A cheap=5, new releases=8 x + y = 13 12. 13. 14. 15. 16. 17. 18. 19. 20.
5.00x + 19.50y = 181 C A A B A A A C B 2
We need to find numbers: b and d such that: x + 4x − 77 = (x b)(x d) where b ⋅ d = 77 and b − d = 4 (the difference since the constant coefficient −77 < 0). We find: b = 7 and d = 11. Since the constant coefficient is −77 < 0, the signs in front of b and d are different. Since the x coefficient is 4 > 0, the sign in front of the larger number 11 is positive. 2
We have: x + 4x − 77 = (x − 7)(x + 11).
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ID: A SHORT ANSWER
21.
ÁÊË −3, 3 ˜ˆ¯
22.
ÊÁ 3, 5 ˆ˜ Ë ¯ 23. (–3, 0)
24. c = 15 + 0.11m c = 18 + 0.08m 100 miles 25. width = 3 cm, length = 19 cm 26. 14, 39 27. 250 28. (–3, –5) 29. h = $3.00, s = $2.00 2 /4
ID: A ÁÊ 1 1 ˜ˆ 30. ÁÁÁÁ , − ˜˜˜˜ ÁË 2 4 ˜¯ 31. 4 32. a. 3x − y = 60 b. x + y = 40 c. (25, 15) 33. a. 2 miles per hour b. 8 miles per hour ÊÁ 10 16 ˆ˜ ˜˜ 34. ÁÁÁÁ − , ˜ ÁË 9 9 ˜˜¯ 35. (–1, –2) 36. (1, –4) 37. The error is in the use of the distributive property in the second line of the solution. Due to the subtraction, −2 must be distributed over the quantity ( 2x − 2 ) and the next line must be 5x − 4x + 4 = 18. The remaining steps are: x + 4 = 18 x = 14 x = 14 y = 2 ( 14 ) − 2 y = 28 − 2 = 26 Solution: x = 14 and y = 26 38. Yes 39. y = –3x + 1 y = −3x + 1 infinite number of solutions 40. 60x – 40x
2
2
15x + x − 28 ( x− 9 ) ( x− 7 ) (x − 4)(x − 3) ( 5x − 1 ) ( 5x − 2 ) (5x+ 3)(x+ 8) ( 2x+ 1 ) ( 2x− 3 ) x = −8, x = 1 9, 7 −2, 8 −9, 3 6 5 51. − , 5 6 1 52. –2, 4 41. 42. 43. 44. 45. 46. 47. 48. 49. 50.
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ID: A 5 53. x = 3 (Reject solution of x = − .) 2 54. a. 4 cm and 6 cm; Write and solve an area equation. x ( x + 2 ) = 24 2
x + 2x − 24 = 0 (x + 6 ) (x − 4 ) = 0 So, x = –6 or x = 4. A side length cannot be negative, so choose x = 4. The side lengths are 4 cm and 4 + 2 = 6 cm. b. The perimeter is 2 ( 4 + 6 ) = 20 centimeters. 55. a. t = 5 or t = –5.5 b. yes; Equivalent equations have the same solution. The solutions found in part (a) are also solutions of the new equation.
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