# Exercises: Introduction To the Language of Algebra (1.2)

## Exercises: Use Variables and Algebraic Symbols

Instructions: For questions 1-14, translate from algebra to English.

1. $16-9$

Solution

$16$ minus $9$, the difference of sixteen and nine

2. $3\cdot9$

3. $28\div4$

Solution

$28$ divided by $4$, the quotient of twenty-eight and four

4. $x+11$

5. $(2)(7)$

Solution

$2$ times $7$, the product of two and seven

6. $(4)(8)$

7. $14<21$

Solution

fourteen is less than twenty-one

8. $17<35$

9. $36\ge 19$

Solution

thirty-six is greater than or equal to nineteen

10. $6n=36$

11. $y-1>6$

Solution

$y$ minus $1$ is greater than $6$, the difference of $y$ and one is greater than six

12. $y-4>8$

13. $2\le 18\div6$

Solution

$2$ is less than or equal to $18$ divided by $6$; two is less than or equal to the quotient of eighteen and six

14. $a\ne 1\cdot12$

## Exercises: Expression or Equation

Instructions: For questions 15-20, determine if each is an expression or an equation.

15. $9\cdot6=54$

Solution

equation

16. $7\cdot 9=63$

17. $5\cdot 4+3$

Solution

expression

18. $x+7$

19. $x+9$

Solution

expression

20. $y-5=25$

## Exercises: Simplify Expressions Using the Order of Operations

Instructions: For questions 21-24, simplify each expression.

21. $5^3$

Solution

$125$

22. $8^3$

23. $2^8$

Solution

$256$

24. $10^5$

## Exercises: Simplify Using Order of Operations

Instructions: For questions 25-46, simplify using the order of operations.

25. $3+8\cdot5$

Solution

$43$

26. $2+6\cdot3$

27. $(3+8)\cdot5$

Solution

$55$

28. $(2+6)\cdot3$

29. $2^3-12\div(9-5)$

Solution

$5$

30. $3^2-18\div(11-5)$

31. $3\cdot8+5\cdot2$

Solution

$34$

32. $4\cdot7+3\cdot5$

33. $2+8(6+1)$

Solution

$58$

34. $4+6(3+6)$

35. $4\cdot12/8$

Solution

$6$

36. $2\cdot36/6$

37. $(6+10)\div(2+2)$

Solution

$4$

38. $(9+12)\div(3+4)$

39. $20\div4+6\cdot5$

Solution

$35$

40. $33\div3+8\cdot2$

41. $3^2+7^2$

Solution

$58$

42. $(3+7)^2$

43. $3(1+9\cdot6)-4^2$

Solution

$149$

44. $5(2+8\cdot4)-7^2$

45. $2\left[1+3(10-2)\right]$

Solution

$50$

46. $5\left[2+4(3-2)\right]$

## Exercises: Evaluate an Expression

Instructions: For questions 47-60, evaluate the expressions.

47. $7x+8$ when $x=2$

Solution

$22$

48. $8x-6$ when $x=7$

49. $x^2$ when $x=12$

Solution

$144$

50. $x^3$ when $x=5$

51. $x^5$ when $x=2$

Solution

$32$

52. $4^x$ when $x=2$

53. $x^2+3x-7$ when $x=4$

Solution

$21$

54. $6x+3y-9$ when
$x=6,y=9$

55. $(x-y)^2$ when
$x=10,y=7$

Solution

$9$

56. $(x+y)^2$ when
$x=6,y=9$

57. $a^2+b^2$ when
$a=3,b=8$

Solution

$73$

58. $r^2-s^2$ when
$r=12,s=5$

59. $2l+2w$ when
$l=15,w=12$

Solution

$54$

60. $2l+2w$ when
$l=18,w=14$

## Exercises: Simplify Expressions by Combining Like Terms

Instructions: For questions 61-64, identify the coefficient of each term.

61. $8a$

Solution

$8$

62. $13m$

63. $5r^2$

Solution

$5$

64. $6x^3$

## Exercises: Identify Like Terms

Instructions: For questions 65-68, identify the like terms.

65. $x^3,8x,14,8y,5,8x^3$

Solution

$x^3\text{ and }8x^3,14\text{ and }5$

66. $6z,3w^2,1,6z^2,4z,w^2$

67. $9a,a^2,16,16b^2,4,9b^2$

Solution

$16\text{ and }4,16b^2\text{ and }9b^2$

68. $3,25r^2,10s,10r,4r^2,3s$

## Exercises: Identify Terms in an Expression

Instructions: For questions 69-72, identify the terms in each expression.

69. $15x^2+6x+2$

Solution

$15x^2,6x,2$

70. $11x^2+8x+5$

71. $10y^3+y+2$

Solution

$10y^3,y,2$

72. $9y^3+y+5$

## Exercises: Simplify by Combining Like Terms

Instructions: For questions 73-82, simplify the expressions by combining like terms.

73. $10x+3x$

Solution

$13x$

74. $15x+4x$

75. $4c+2c+c$

Solution

$7c$

76. $6y+4y+y$

77. $7u+2+3u+1$

Solution

$10u+3$

78. $8d+6+2d+5$

79. $10a+7+5a-2+7a-4$

Solution

$22a+1$

80. $7c+4+6c-3+9c-1$

81. $3x^2+12x+11+14x^2+8x+5$

Solution

$17x^2+20x+16$

82. $5b^2+9b+10+2b^2+3b-4$

## Exercises: Translate an English Phrase to an Algebraic Expression

Instructions: For questions 83-94, translate the phrases into algebraic expressions.

83. the difference of $14$ and $9$

Solution

$14-9$

84. the difference of $19$ and $8$

85. the product of $9$ and $7$

Solution

$9\cdot7$

86. the product of $8$ and $7$

87. the quotient of $36$ and $9$

Solution

$36\div9$

88. the quotient of $42$ and $7$

89. the sum of $8x$ and $3x$

Solution

$8x+3x$

90. the sum of $13x$ and $3x$

91. the quotient of $y$ and $3$

Solution

$\frac{y}{3}$

92. the quotient of $y$ and $8$

93. eight times the difference of $y$ and nine

Solution

$8(y-9)$

94. seven times the difference of $y$ and one

## Exercises: Word Problems

Instructions: For questions 95-98, write an expression for the given word problems.

95. Eric has rock and classical CDs in his car. The number of rock CDs is 3 more than the number of classical CDs. Let $c$ represent the number of classical CDs. Write an expression for the number of rock CDs.

Solution

$c+3$

96. The number of girls in a second-grade class is 4 less than the number of boys. Let $b$ represent the number of boys. Write an expression for the number of girls.

97. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let $n$ represent the number of nickels. Write an expression for the number of pennies.

Solution

$2n-7$

98. Jeannette has $5$ and $10$ bills in her wallet. The number of fives is three more than six times the number of tens. Let $t$ represent the number of tens. Write an expression for the number of fives.

## Exercises: Everyday Math

Instructions: For questions 99-100, answer the given everyday math word problems.

99. Car insurance. Justin’s car insurance has a $750$ deductible per incident. This means that he pays $750$ and his insurance company will pay all costs beyond $750$ . If Justin files a claim for $2,100$:

a. how much will he pay?
b. how much will his insurance company pay?

Solution

a. $750$
b. $1,350$

100. Home insurance. Armando’s home insurance has a $2,500$ deductible per incident. This means that he pays $2,500$ and the insurance company will pay all costs beyond $2,500$. If Armando files a claim for $19,400$:

a. how much will he pay?
b. how much will the insurance company pay?

## Exercises: Writing Exercises

Instructions: For questions 101-104, answer the given writing exercises.

101. Explain the difference between an expression and an equation.

Solution

102. Why is it important to use the order of operations to simplify an expression?

103. Explain how you identify the like terms in the expression $8a^2+4a+9-a^2-1$.

Solution

104. Explain the difference between the phrases “4 times the sum of $x$ and $y$” and “the sum of 4 times $x$ and $y$.”