variables & expressions chapter...
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Variables & Expressions Chapter Questions
1. What is the difference between an expression and an equation?
2. Can you name 3 words that indicate each operation?
3. How do you evaluate an expression?
4. Explain how distribution can simplify a problem.
5. What are like terms?
6. How do you combine like terms?
Variables & Expressions Chapter Problems
Vocabulary, Equations & Expressions
Classwork
1. Circle the constant and underline the coefficient for each expression below
a. 5x – 3
b. 2x + 7
c. 2 – 4x
d. x + 3
2. Create an algebraic expression with a coefficient of 7 and a constant of 4.
3. Create an algebraic expression with a coefficient of -1 and a constant of -12.
4. Create an equation that contains a coefficient of 6.
5. Create an equation that contains a coefficient of -13.
6. What is the difference between an algebraic expression and an equation?
7. Which are algebraic expressions?
5x – 2 8x w 14 + 5x 2w – 6 4x – 8 = 9
Homework
8. Circle the constant and underline the coefficient for each expression below
a. 3x – 5
b. 2x - 1
c. 7 – 8x
d. x + 2
9. Create an algebraic expression with a coefficient of 17 and a constant of 3.
10. Create an algebraic expression with a coefficient of -1 and a constant of -1.
11. Create an equation that contains a coefficient of 4.
12. Create an equation that contains a constant of -12.
13. What is the difference between an algebraic expression and an equation?
14. Which are algebraic expressions?
17m 8 – 3z w 9w + 4 = 12 12 + 7t 6y + 4
Translating between Words & Expressions
Classwork
Translate the words into an algebraic expression.
15. 4 times x
16. The sum of x and 6
17. The product of 9 and y
18. w less than 8
19. 5 more than x
20. The difference of 6 and x
21. 9 times the product of x and 4
22. The product of 5 and y divided by 3
23. The quotient of 300 and the quantity of x times 2
24. x less than 32
25. The quotient of 35 and the quantity of x minus 7
26. The product of 7 and x minus the quantity of 4 less than y
27. The quantity of 9 more than x divided by the quantity of 12 less than y
Homework
Translate the words into an algebraic expression.
28. The product of 14 and x
29. The quotient of x and 5
30. The sum of 19 and w
31. w less than 8
32. 7 less than x
33. The difference of 16 and y
34. 9 times the quotient of x and 20
35. The product of 6 and x less 3
36. The quotient of 100 and the sum of x and 2
37. x less than 2
38. The product of 5 and the quantity of x less than 7
39. The product of 27 and y divided by the quantity of 4 more than y
40. The quantity of 6 less than x divided by the quantity of 2 more than y
Tables & Expressions
Classwork Complete the table. 41.
n 3n
5
10
15
42. 43. 44. 45. 46.
47. Adult ticket prices are $3 more than child ticket prices. Determine the adult ticket price, given the child
ticket price.
Child Ticket Price Adult Ticket Price
$5
$7
$10
$12
48. Write an expression that represents the adult price, if the child price is “x” 49. For NJASK testing, 25 students are placed in each classroom. Determine the number of classrooms
needed, given the number of students testing.
n n + 7
3
5
7
n n - 70
80
100
120
140
n n ÷ 8
0
1
8
16
n 4 less than n
20
18
16
14
n 2 more than n
20
18
16
14
Number of Students Testing Number of Classroom Needed
250
325
400
520
50. Write an expression that represents the number of classrooms needed, if the number of students testing is “x”
51. Mary has ½ the amount of money that Jim has. Determine the amount of money that Mary has, given Jim’s amount of money.
52. Write an expression that represents the amount of money Mary has, given the amount of Jim’s money.
53. Each person running in the race paid $20. Determine the amount of money collected, given the amount of people running in the race.
54. Write an expression that represents the amount of money collected, given the number of people running in the race.
Write an expression for the following situations. 55. Bob weighs 7 more pounds than Jack. Jack weighs x pounds. Bob’s weight: 56. Tiffany has 6 dollars less than Jessica. Jessica has x dollars. Tiffany’s money: 57. Samantha has 12 more stickers than Mike. Mike has S stickers. Samantha’s sticker amount: 58. The recipe calls for twice the amount of sugar than flour. There is F amount of flour in the recipe.
Amount of sugar: 59. Mark’s quiz grade is one more than twice Ted’s quiz grade. Ted’s quiz grade is x. Mark’s quiz grade: 60. Laura paid x dollars for her prom dress. Beth paid four dollars less than Laura. Beth’s prom gown
price: 61. David ran the 5k in x minutes. Harry ran the same race in five minutes less than double David’s time.
Harry’s time: 62. The beans grew K inches. The tomatoes grew 3 inches more than triple the height of the beans.
Tomato height: Create a scenario for the following expressions: 63. x + 5 64. 2(x – 3)
Jim’s amount of money Mary’s amount of money
$50
$100
$175
$220
Number of People Running Amount of Money Collected
150
230
410
520
Homework
Complete the table.
65.
66.
67.
68.
69.
70.
71. Child ticket prices are $3 less than adult ticket prices. Determine the child ticket price, given the adult
ticket price.
n 5 + n
5
10
15
n 7n
3
5
7
n n 10
80
100
120
140
n n ÷ 2
0
1
8
16
n 34 less n
20
18
16
14
n 5 less than n
20
18
16
14
Adult Ticket Price Child Ticket Price
$10
$15
$20
$25
72. Write an expression that represents the child price, if the adult price is “x”
73. For bussing, 40 students are assigned to each bus. Determine the number of busses needed, given
the number of students riding.
Number of Students Riding Number of Busses Needed
240
320
400
500
74. Write an expression that represents the number of busses needed, if the number of students riding is
“x”
75. The farm always has four times the number of chicks as hens. Determine the number of chicks,
given the number of hens.
Number of hens Number of chicks
20
40
50
60
76. Write an expression that represents the number of chicks, given the number of hens.
77. Each person running in the race will eat two hotdogs. Determine the number of hotdogs needed, given the amount of people running in the race.
Number of People Running Number of Hotdogs needed
150
230
410
520
78. Write an expression that represents the number of hotdogs needed, given the number of people
running in the race.
Write an expression for the following situations.
79. Bob weighs 17 pounds less than Jack. Jack weighs x pounds. Bob’s weight:
80. Tiffany has 50 dollars more than Jessica. Jessica has x dollars. Tiffany’s money:
81. Samantha has 12 times as many stickers than Mike. Mike has S stickers. Samantha’s sticker amount:
82. The recipe calls for triple the amount of sugar than flour. There is F amount of flour in the recipe.
Amount of sugar:
83. Mark’s quiz grade is six more than double Ted’s quiz grade. Ted’s quiz grade is x. Mark’s quiz grade:
84. Laura paid x dollars for her prom dress. Beth paid 16 dollars more than Laura.
Beth’s prom gown price:
85. David ran the 5k in x minutes. Harry ran the same race in half the time that David ran the race.
Harry’s time:
86. The beans grew K inches. The tomatoes grew triple the height of the beans less 2 inches. Tomato
height:
Create a scenario for the following expressions:
87. 2(x + 3)
88. x - 4
Evaluating Expressions
Classwork
89. Evaluate the expression for the given value
a. (2n + 1)2 for n = 3 b. 2(n + 1)2 for n = 4 c. 2n + 22 for n = 3 d. 4x + 3x for x = 5 e. 3(x – 3) for x = 7 f. 8(x + 5)(x – 2) for x = 4 g. 3x2 for x = 2 h. 5x + 45 for x = 6 i. 4x for x = 10
5 j. 4y + x for x = 2 and y = 3 k. x + 17 for x = 12 and y = ½
y l. 6x + 8y for x = 9 and y = ¼ m. x + (2x – 8) for x = 10 n. 5(3x) + 8y for x = 2 and y = 10
90. Use the distance formula, D = rt, to find the distance traveled
a. Rate: 40 mph; Time: 2 hrs b. Rate: 60 mph; Time: 5 hrs c. Rate: 34 mph; Time: ½ hr
Homework
91. Evaluate the expression for the given value
a. (2n + 1)2 for n = 1 b. 2(n + 1)2 for n = 3 c. 2n + 22 for n = 5 d. 4x + 3x for x = 6 e. 3(x – 3) for x = 3 f. 8(x + 5)(x – 2) for x = 6 g. 3x2 for x = 8 h. 5x + 45 for x = 3 i. 4x for x = 15 5 j. 4y + x for x = 12 and y = 13 k. x + 17 for x = 2 and y = ½
y l. 6x + 8y for x = 8 and y = ¾ m. x + (2x – 8) for x = 11 n. 5(3x) + 8y for x = 12 and y = 5
92. Use the distance formula, D = rt, to find the distance traveled
a. Rate: 14 mph; Time: 2 hrs b. Rate: 60 mph; Time: ¾ hrs c. Rate: 40 mph; Time: ½ hr
Distributive Property
Classwork
93. Use the Distributive Property to rewrite the expressions without parentheses
a. (x + 4) b. 8(x – 2) c. 6(x + 4) d. -1(x – 4) e. (x + 2)8
Homework
94. Use the Distributive Property to rewrite the expressions without parentheses
a. 5(x + 4) b. 7(x – 12) c. 3(x - 14) d. -1(x – 2) e. (x - 2)5
Like Terms
Classwork 95. Create a like term for the given term.
a. 4x b. 13y c. 15x2
d. 16xy e. X
Homework 96. Create a like term for the given term.
a. 6x b. Y c. 10x2 d. 14xy e. -5x
Combining Like Terms
Classwork
97. Simplify the expression if possible.
a. 7x + 8x b. 6x + 8y + 2x c. 15x2 + 5x2 d. 5x +2(x + 8) e. -10y + 4y f. 9(x + 5) + 7(x – 3) g. 8 + (x – 4)2 h. 7y + 8x + 3y + 2x i. x + 2x j. x2 + 5x2 k. 2x + 4x + 3 l. 6y – 3y m. 9y + 4y – 2y + y n. x + 5x + x + 12 o. 8x – 3x + 2x + 15
Homework
98. Simplify the expression if possible.
a. 17x + 18x + 3 b. 6x + 8y - 2x – y c. 15x2 + 5x2 + 2x d. 5x +2(x + 8) + 3 e. -10y + 4y – 5 f. 9(x - 5) + 7(x + 3) g. 18 + (x – 4)2 – 4 h. 7y + 8x + 3y + 2x + 9 i. x + 2x + x + 5x j. 6x2 + 5x2 k. 12x + 14x + 3y l. 6y – 3y + 6xy + 4xy m. 9y + 4y – 2y + y + y2 n. x + 5x + x + 12 – 7x o. 8x – 3x + 2x + 15 – 7y
Variables & Expressions Multiple Choice Questions
Determine whether the given terms are like terms. Circle your response.
1. 3x and -2x Are Like Terms Are Unlike Terms
2. 5a and 5b Are Like Terms Are Unlike Terms
3. 4y and 5xy Are Like Terms Are Unlike Terms
4. x2y and xy2 Are Like Terms Are Unlike Terms
5. 22 and 14 Are Like Terms Are Unlike Terms
6. xy and –xy Are Like Terms Are Unlike Terms
7. Match the expression 3(-4 + 3) with an equivalent expression.
a) 4(3) + 4(3) b) 3(-4) + 3(3)
c) 4(3) - 4(3) d) 3(4) + 3(3)
8. Which algebraic expression represents the number of days in w weeks?
a) w – 7 b) w/7
c) w + 7 d) 7w
9. Which algebraic expression represents the number of hours in m minutes?
a) m – 60 b) m/60
c) m + 60 d) 60m
10. In the expression 3x + 5, the value of 3 best describes:
a) the constant b) the operation
c) the variable d) the coefficient
11. In the expression 2x + 16, the value of 16 best describes:
a) the coefficient b) the variable
c) the operation d) the constant
12. Evaluate the expression 2x, when x = 10
a) 20 b) 12
c) 210 d) 1
5
13. What operation is being performed between the coefficient and variable in the expression20
x?
a) addition b) division
c) subtraction d) multiplication
14. A group of 15 parents buys tickets to a fundraiser show and receives a group discount of $2 off
the regular ticket price p. Which expression represents the total cost of the tickets, in dollars?
a) 15 • p + 2 b) 15 • (p - 2)
c) p - 15 • 2 d) p • (15 - 2)
15. A music store sells CDs for $15 and tapes for $3. Which expression could be used to find the
dollar total of the sales for an hour if the store sold 8 CDs and 5 tapes?
a) (8 + 15) • (5 + 3) b) (8 •15) + (5 • 3)
c) (8 • 3) + (5 •15) d) (15 ÷8) + (5 ÷ 3)
16. There were three times as many adults as students attending a school play. If the attendance
was 480, how many adults and how many students attended the play?
a) 360 students b) 240 students
120 adults 240 adults
c) 120 students d) 160 students
360 adults 320 adults
17. Which of the following is not a variable expression?
a) 4n b) n + m
c) n - 4 d) 4 + 3
18. What is the value of the expression x + y when x = 15 and y = 21?
a) 6 b) 30
c) 36 d) 42
19. Evaluate n
2 - m when m = 7 and n = 8
a) 9 b) -9
c) 57 d) 71
20. Claire has had her driver’s license for three years. Bill has had his license for “b” fewer years
than Claire. Which expression can be used to show the number of years Bill has had his driver’s
license?
a) 3 + b b) b + 3
c) 3 - b d) b < 3
21. Which situation is best modeled by the expression 25 – x?
a) George places “x” more video games on a shelf with 25 games
b) Sarah has driven “x” miles of a 25 mile trip
c) Ameilia paid $25 of an “x” dollar lunch she shared with Ariel
d) George has 25 boxes full of “x” baseball cards each
22. Evaluate -3x + 5 when x = -2
a) 11 b) -1
c) 1 d) -11
23. Nine decreased by the quantity eight times a number “x”.
a) 8x - 9 b) 9 – 8x
c) 9x - 8 d) 8 – 9x
24. Four more than the quotient of 25 and y.
a) 25
4y
b) y
254
c) 𝟐𝟓+𝟒
𝒚 d)
𝒚
𝟐𝟓−𝟒
25. What is the coefficient of x in the expression 4y + 5 - x?
a) 5 b) 1
c) -1 d) 0
Variables & Expressions Short Constructed Response
1. A rectangle is 6 inches longer than it is wide. Write and simplify an expression for the perimeter
of the rectangle in terms of the width w.
2. You and a friend worked in the school store last week. You worked 4 hours less than your friend.
Let h be the number of hours your friend worked. Write an expression in simplest form that
represents the total number of hours you both worked.
3. A trail mix contains peanuts, raisins, and M&Ms. In the mix, the amount of peanuts is three
times the amount of M&Ms; and the amount of raisins is two times the amount of M&Ms. Let m
represent the amount of M&Ms. Write and simplify an expression for the total number of pieces
of food in the trail mix.
4. Write an expression containing three terms that is in simplest form. One of the terms should be
a constant.
5. Simplify: 5 – 2(3x – 4) + x
6. Shelly lives 500 miles away. Paul drove 65 m.p.h. for 4 hours. How many more miles will it take
for him to arrive at Shelly’s house.
7. Evaluate the expression
5
9 (F – 32) when F = 41
Variables & Expressions Extended Constructed Response
1. At the video arcade, Jenny buys 25 tokens. She uses two tokens for each game she plays.
a) Write an expression for the number of tokens Jenny has left after playing g games.
b) Find the number of tokens Jenny has left after playing 1, 4, 6, 10 and 12 games.
2. Bob wants to go to the movies with his friends. The movie theater charges $8 per ticket. Bob’s
friends reserve $48.00 worth of tickets in advance. How many people in total can attend the
movie?
a) Identify the variable
b) Identify the constant
c) Write an equation which includes the number of people attending the movie, the price
of each ticket, and the total cost of the movie.
3. Write an expression that has four terms and simplifies to 16x + 5.
a) Identify the like terms
b) Identify the coefficients
c) Identify the constant terms
4. Mary is 5 years older than Bob. If Bob lives to be 65, 70, and 75 years of age, what will Mary’s
age be at the same time? Complete the chart with an expression containing a variable to
explain your answer.
Bob Mary
5. A cell phone company is offering 2 different monthly plans. Each plan charges a monthly fee
plus an additional cost per minute.
Plan A: $ 40 fee plus $0.45 per minute
Plan B: $70 fee plus $0.35 per minute
a) Write an expression to represent the cost of Plan A
b) Write an expression to represent the cost of Plan B
c) Which plan would be least expensive for a total of 100 minutes?
6. Chad complained to his friend that he had five equations to solve for homework.
Are all of the homework problems equations? Justify your answer.
Math Homework
1. 3x2 ∙ 2x4
2. 5 – 2x = 3x 3 3(2x + 7) 4. 7x2 + 2x – 3x2 – 9
5. 2 = x + 2 3 6
From the New York State Education Department. Office of Assessment Policy, Development and
Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.
Answer Key
1. a. constant: -3, coefficient: 5 b. constant: 7, coefficient: 2 c. constant: 2, coefficient: -4 d. constant: 3, coefficient: 1
2. 7x+4 3. –x-12 4. 6x+1 5. -13x+1 6. An algebraic expression is unsolvable and does not contain an equal sign. 7. 5x-2, 8x, w, 14+5x, 2w-6 8.
a. constant: -5, coefficient: 3 b. constant: -1, coefficient: 2 c. constant: 7, coefficient: -8 d. constant: 2, coefficient: 1
9. 17x+3 10. –x-1 11. 4x+2 12. -12x+2 13. An algebraic expression is unsolvable and does not contain an equal sign. 14. 17m, 8-3z, w 12+7t, 6y+4 15. 4x 16. x+6 17. 9y 18. 8-w 19. 5+x 20. 6-x 21. 9(x+4) 22. 5y/3 23. 300/2x 24. 32-x 25. 35/(x-7) 26. 7x-(y-4) 27. 9+(x/(y-12)) 28. 14x 29. x/5 30. 19+w 31. 8-w 32. x-7 33. 16y 34. 9x/20 35. 6(3-x) 36. 100/(x+2) 37. 2-x 38. 5(7-x) 39. 27y/(4+y) 40. (x-6)/(2+y) 41.
n 3n
5 15
10 30
15 45
42.
43.
44.
45.
46.
47.
Child Ticket Price Adult Ticket Price
$5 $8
$7 $10
$10 $13
$12 $15
48. x+3 49.
n n + 7
3 10
5 12
7 14
n n - 70
80 10
100 30
120 50
140 70
n n ÷ 8
0 0
1 1/8
8 1
16 2
n 4 less than n
20 16
18 14
16 12
14 10
n 2 more than n
20 22
18 20
16 18
14 16
Number of Students Testing Number of Classroom Needed
250 10
325 13
400 16
520 21
50. x/25
51.
52. x/2 53.
54. 20x 55. x+7 56. x-6 57. s+12 58. 2f 59. 2x+1 60. x-4 61. 2x-5 62. 3k+3 63. Multiple Answers 64. Multiple Answers 65.
66.
67.
Jim’s amount of money Mary’s amount of money
$50 $25
$100 $50
$175 $87.50
$220 $110
Number of People Running Amount of Money Collected
150 $3000
230 $4600
410 $8200
520 $10400
n 5+n
3 10
5 15
7 20
n 7n
3 21
5 35
7 49
n n 10
68.
69.
70.
71.
Adult Ticket Price Child Ticket Price
$10 $7
$15 $12
$20 $17
$25 $22
72. x-3 73.
Number of Students Riding Number of Busses Needed
80 8
100 10
120 12
140 8
n n ÷ 2
0 0
1 .5
8 4
16 8
n 34 less n
20 -14
18 -16
16 -18
14 -20
n 5 less than n
20 15
18 13
16 11
14 9
240 6
320 8
400 10
500 13
74. x/40 75.
Number of hens Number of chicks
20 80
40 160
50 200
60 240
76. 4x
77.
Number of People Running Number of Hotdogs needed
150 300
230 460
410 820
520 1040
78. 2x 79. x-17 80. 50+x 81. 12S 82. 3F 83. 6+2x 84. x+16 85. x/2 86. 3(K-2) 87. Multiple Answers 88. Multiple Answers 89.
a. 49 b. 50 c. 10 d. 35 e. 12 f. 144 g. 12 h. 75 i. 8 j. 14 k. 41 l. 56 m. 22 n. 110
90.
a. 80 b. 300 c. 17
91. a. 9 b. 32 c. 14 d. 42 e. 0 f. 352 g. 192 h. 60 i. 12 j. 64 k. 21 l. 54 m. 25 n. 220
92. a. 28 b. 45 c. 20
93. a. x+4 b. 8x-16 c. 6x+24 d. –x+4 e. 8x+16
94. a. 5x+20 b. 7x-84 c. 3x-42 d. –x+2 e. 5x-10
95. a. Multiple Answers ex:2(2x) b. Multiple Answers ex:26y/2 c. Multiple Answers ex:(3x)(5x) d. Multiple Answers (4x)(4y) e. Multiple Answers ex:x2/x
96. a. Multiple Answers b. Multiple Answers c. Multiple Answers d. Multiple Answers e. Multiple Answers
97. a. 15x b. 8x+8y c. 20x2 d. 7x+16 e. -6y f. 16x+24 g. 2x h. 10y+10x i. 3x
j. 6x2 k. 6x+3 l. 3y m. 12y n. 7x+12 o. 7x+15
98. a. 35x+3 b. 4x+7y c. 20x2+2x d. 7x+19 e. -6y-5 f. 16x-24 g. 2x+6 h. 10y+10x+9 i. 9x j. 11x2 k. 26x+3y l. 3y+10xy m. 12y+y2 n. 12 o. 7x+15-7y