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Study Guides > ALGEBRA / TRIG I

Problem Set: The Language of Algebra

Using the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.
  1. 16916 - 9

    Answer: 16 minus 9, the difference of sixteen and nine

  2. 25725 - 7
  3. 565\cdot 6

    Answer: 5 times 6, the product of five and six

  4. 393\cdot 9
  5. 28÷428\div 4

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

  6. 45÷545\div 5
  7. x+8x+8

    Answer: x plus 8, the sum of x and eight

  8. x+11x+11
  9. (2)(7)\left(2\right)\left(7\right)

    Answer: 2 times 7, the product of two and seven

  10. (4)(8)\left(4\right)\left(8\right)
  11. 14<2114<21

    Answer: fourteen is less than twenty-one

  12. 17<3517<35
  13. 361936\ge 19

    Answer: thirty-six is greater than or equal to nineteen

  14. 422742\ge 27
  15. 3n=243n=24

    Answer: 3 times n equals 24, the product of three and n equals twenty-four

  16. 6n=366n=36
  17. y1>6y - 1>6

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

  18. y4>8y - 4>8
  19. 218÷62\le 18\div 6

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

  20. 320÷43\le 20\div 4
  21. a74a\ne 7\cdot 4

    Answer: a is not equal to 7 times 4, a is not equal to the product of seven and four

  22. a112a\ne 1\cdot 12

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.
  1. 96=549\cdot 6=54

    Answer: equation

  2. 79=637\cdot 9=63
  3. 54+35\cdot 4+3

    Answer: expression

  4. 63+56\cdot 3+5
  5. x+7x+7

    Answer: expression

  6. x+9x+9
  7. y5=25y - 5=25

    Answer: equation

  8. y8=32y - 8=32

Simplify Expressions with Exponents

In the following exercises, write in exponential form.
  1. 33333333\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3

    Answer: 37

  2. 4444444\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4
  3. xxxxxx\cdot x\cdot x\cdot x\cdot x

    Answer: x5

  4. yyyyyyy\cdot y\cdot y\cdot y\cdot y\cdot y

Simplify Expressions with Exponents

In the following exercises, write in expanded form.
  1. 53{5}^{3}

    Answer: 125

  2. 83{8}^{3}
  3. 28{2}^{8}

    Answer: 256

  4. 105{10}^{5}

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.
  1. 3+853+8\cdot 5

    Answer: 43

  2. (3+8)5\text{(3+8)}\cdot \text{5}

    Answer: 55

  3. 2+632+6\cdot 3
  4. (2+6)3\text{(2+6)}\cdot \text{3}
  5. 2312÷(95){2}^{3}-12\div \left(9 - 5\right)

    Answer: 5

  6. 3218÷(115){3}^{2}-18\div \left(11 - 5\right)
  7. 38+523\cdot 8+5\cdot 2

    Answer: 34

  8. 47+354\cdot 7+3\cdot 5
  9. 2+8(6+1)2+8\left(6+1\right)

    Answer: 58

  10. 4+6(3+6)4+6\left(3+6\right)
  11. 412/84\cdot 12/8

    Answer: 6

  12. 236/62\cdot 36/6
  13. 6+10/2+26+10/2+2

    Answer: 13

  14. 9+12/3+49+12/3+4
  15. (6+10)÷(2+2)\left(6+10\right)\div \left(2+2\right)

    Answer: 4

  16. (9+12)÷(3+4)\left(9+12\right)\div \left(3+4\right)
  17. 20÷4+6520\div 4+6\cdot 5

    Answer: 35

  18. 33÷3+8233\div 3+8\cdot 2
  19. 20÷(4+6)520\div \left(4+6\right)\cdot 5

    Answer: 10

  20. 33÷(3+8)233\div \left(3+8\right)\cdot 2
  21. 42+52{4}^{2}+{5}^{2}

    Answer: 41

  22. 32+72{3}^{2}+{7}^{2}
  23. (4+5)2{\left(4+5\right)}^{2}

    Answer: 81

  24. (3+7)2{\left(3+7\right)}^{2}
  25. 3(1+96)423\left(1+9\cdot 6\right)-{4}^{2}

    Answer: 149

  26. 5(2+84)725\left(2+8\cdot 4\right)-{7}^{2}
  27. 2[1+3(102)]2\left[1+3\left(10 - 2\right)\right]

    Answer: 50

  28. 5[2+4(32)]5\left[2+4\left(3 - 2\right)\right]

Everyday Math

Basketball

In the 2014 NBA playoffs, the San Antonio Spurs beat the Miami Heat. The table below shows the heights of the starters on each team. Use this table to fill in the appropriate symbol (=,<,>)\text{(=},\text{<},\text{>)}.
Spurs Height Heat Height
Tim Duncan 83" Rashard Lewis 82"
Boris Diaw 80" LeBron James 80"
Kawhi Leonard 79" Chris Bosh 83"
Tony Parker 74" Dwyane Wade 76"
Danny Green 78" Ray Allen 77"
  1. Height of Tim Duncan____Height of Rashard Lewis
  2. Height of Boris Diaw____Height of LeBron James
  3. Height of Kawhi Leonard____Height of Chris Bosh
  4. Height of Tony Parker____Height of Dwyane Wade
  5. Height of Danny Green____Height of Ray Allen

Elevation

In Colorado there are more than 5050 mountains with an elevation of over 14,000 feet.14,000\text{ feet.} The table shows the ten tallest. Use this table to fill in the appropriate inequality symbol.
Mountain Elevation
Mt. Elbert 14,433'
Mt. Massive 14,421'
Mt. Harvard 14,420'
Blanca Peak 14,345'
La Plata Peak 14,336'
Uncompahgre Peak 14,309'
Crestone Peak 14,294'
Mt. Lincoln 14,286'
Grays Peak 14,270'
Mt. Antero 14,269'
Elevation of La Plata Peak____Elevation of Mt. Antero Elevation of Blanca Peak____Elevation of Mt. Elbert Elevation of Gray’s Peak____Elevation of Mt. Lincoln Elevation of Mt. Massive____Elevation of Crestone Peak Elevation of Mt. Harvard____Elevation of Uncompahgre Peak

Writing Exercises

Explain the difference between an expression and an equation. [practice-area rows="4"][/practice-area] Why is it important to use the order of operations to simplify an expression? [practice-area rows="4"][/practice-area]

Evaluating, Simplifying, and Translating Algebraic Expressions

Evaluate Algebraic Expressions

In the following exercises, evaluate the expression for the given value.
  1. 7x+8 when x=27x+8\text{ when }x=2

    Answer: 22

  2. 9x+7 when x=39x+7\text{ when }x=3
  3. 5x4 when x=65x - 4\text{ when }x=6

    Answer: 26

  4.  8x6 when x=78x - 6\text{ when }x=7
  5. x2 when x=12{x}^{2}\text{ when }x=12

    Answer: 144

  6. x3 when x=5{x}^{3}\text{ when }x=5
  7. x5 when x=2{x}^{5}\text{ when }x=2

    Answer: 32

  8. x4 when x=3{x}^{4}\text{ when }x=3
  9. 3x when x=3{3}^{x}\text{ when }x=3

    Answer: 27

  10. 4x when x=2{4}^{x}\text{ when }x=2
  11. x2+3x7 when x=4{x}^{2}+3x - 7\text{ when }x=4

    Answer: 21

  12. x2+5x8 when x=6{x}^{2}+5x - 8\text{ when }x=6
  13. 2x+4y5 when x=7,y=82x+4y - 5\text{ when }x=7,y=8

    Answer: 41

  14. 6x+3y9 when x=6,y=96x+3y - 9\text{ when }x=6,y=9
  15. (xy)2 when x=10,y=7{\left(x-y\right)}^{2}\text{ when }x=10,y=7

    Answer: 9

  16. (x+y)2 when x=6,y=9{\left(x+y\right)}^{2}\text{ when }x=6,y=9

    Answer: 225

  17. a2+b2 when a=3,b=8{a}^{2}+{b}^{2}\text{ when }a=3,b=8

    Answer: 73

  18. r2s2 when r=12,s=5{r}^{2}-{s}^{2}\text{ when }r=12,s=5
  19. 2l+2w when l=15,w=122l+2w\text{ when }l=15,w=12

    Answer: 54

  20. 2l+2w when l=18,w=142l+2w\text{ when }l=18,w=14

Identify Terms, Coefficients, and Like Terms

In the following exercises, list the terms in the given expression.
  1. 15x2+6x+215{x}^{2}+6x+2

    Answer: 15x2, 6x, 2

  2. 11x2+8x+511{x}^{2}+8x+5
  3. 10y3+y+210{y}^{3}+y+2

    Answer: 10y3, y, 2

  4. 9y3+y+59{y}^{3}+y+5
In the following exercises, identify the coefficient of the given term.
  1. 8a8a

    Answer: 8

  2. 13m13m
  3. 5r25{r}^{2}

    Answer: 5

  4. 6x36{x}^{3}
In the following exercises, identify all sets of like terms.
  1. x3,8x,14,8y,5,8x3{x}^{3},8x,14,8y,5,8{x}^{3}

    Answer: x3, 8x3 and 14, 5

  2. 6z,3w2,1,6z2,4z,w26z,3{w}^{2},1,6{z}^{2},4z,{w}^{2}
  3. 9a,a2,16ab,16b2,4ab,9b29a,{a}^{2},16ab,16{b}^{2},4ab,9{b}^{2}

    Answer: 16ab and 4ab; 16b2 and 9b2

  4. 3,25r2,10s,10r,4r2,3s3,25{r}^{2},10s,10r,4{r}^{2},3s

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the given expression by combining like terms.
  1. 10x+3x10x+3x

    Answer: 13x

  2. 15x+4x15x+4x
  3. 17a+9a17a+9a

    Answer: 26a

  4. 18z+9z18z+9z
  5. 4c+2c+c4c+2c+c

    Answer: 7c

  6. 6y+4y+y6y+4y+y
  7. 9x+3x+89x+3x+8

    Answer: 12x + 8

  8. 8a+5a+98a+5a+9
  9. 7u+2+3u+17u+2+3u+1

    Answer: 10u + 3

  10. 8d+6+2d+58d+6+2d+5
  11. 7p+6+5p+47p+6+5p+4

    Answer: 12p + 10

  12. 8x+7+4x58x+7+4x - 5
  13. 10a+7+5a2+7a410a+7+5a - 2+7a - 4

    Answer: 22a + 1

  14. 7c+4+6c3+9c17c+4+6c - 3+9c - 1
  15. 3x2+12x+11+14x2+8x+53{x}^{2}+12x+11+14{x}^{2}+8x+5

    Answer: 17x2 + 20x + 16

  16. 5b2+9b+10+2b2+3b45{b}^{2}+9b+10+2{b}^{2}+3b - 4

Translate English Phrases into Algebraic Expressions

In the following exercises, translate the given word phrase into an algebraic expression.
  1. The sum of 8 and 12

    Answer: 8 + 12

  2. The sum of 9 and 1
  3. The difference of 14 and 9

    Answer: 14 − 9

  4. 8 less than 19
  5. The product of 9 and 7

    Answer: 9 ⋅ 7

  6. The product of 8 and 7
  7. The quotient of 36 and 9

    Answer: 36 ÷ 9

  8. The quotient of 42 and 7
  9. The difference of xx and 44

    Answer: x − 4

  10. 33 less than xx
  11. The product of 66 and yy

    Answer: 6y

  12. The product of 99 and yy
  13. The sum of 8x8x and 3x3x

    Answer: 8x + 3x

  14. The sum of 13x13x and 3x3x
  15. The quotient of yy and 33

    Answer: y3\Large\frac{y}{3}

  16. The quotient of yy and 88
  17. Eight times the difference of yy and nine

    Answer: 8 (y − 9)

  18. Seven times the difference of yy and one
  19. Five times the sum of xx and yy

    Answer: 5 (x + y)

  20. Nine times five less than twice xx

Translate English Phrases into Algebraic Expressions

In the following exercises, write an algebraic expression.
  1. Adele bought a skirt and a blouse. The skirt cost $15\$15 more than the blouse. Let bb represent the cost of the blouse. Write an expression for the cost of the skirt.

    Answer: b + 15

  2. Eric has rock and classical CDs in his car. The number of rock CDs is 33 more than the number of classical CDs. Let cc represent the number of classical CDs. Write an expression for the number of rock CDs.
  3. The number of girls in a second-grade class is 44 less than the number of boys. Let bb represent the number of boys. Write an expression for the number of girls.

    Answer: b − 4

  4. Marcella has 66 fewer male cousins than female cousins. Let ff represent the number of female cousins. Write an expression for the number of boy cousins.
  5. Greg has nickels and pennies in his pocket. The number of pennies is seven less than twice the number of nickels. Let nn represent the number of nickels. Write an expression for the number of pennies.

    Answer: 2n − 7

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

Everyday Math

In the following exercises, use algebraic expressions to solve the problem.

Car insurance

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

Answer: He will pay $750. His insurance company will pay $1350.

Home insurance

Pam and Armando’s home insurance has a $2,500\$2,500 deductible per incident. This means that they pay $2,500\$2,500 and their insurance company will pay all costs beyond $2,500\$2,500. If Pam and Armando file a claim for $19,400\$19,400, how much will they pay, and how much will their insurance company pay?

Writing Exercises

Explain why "the sum of x and y" is the same as "the sum of y and x," but "the difference of x and y" is not the same as "the difference of y and x." Try substituting two random numbers for xx and yy to help you explain. [practice-area rows="4"][/practice-area] Explain the difference between "44 times the sum of xx and yy" and "the sum of 44 times xx and yy." [practice-area rows="4"][/practice-area]

Subtraction Property of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each given value is a solution to the equation.

Exercise 1

x+13=21x+13=21
  1. x=8x=8

    Answer: yes

  2. x=34x=34

    Answer: no

Exercise 2

y+18=25y+18=25
  1. y=7y=7
  2. y=43y=43

Exercise 3

m4=13m - 4=13
  1. m=9m=9

    Answer: no

  2. m=17m=17

    Answer: yes

Exercise 4

n9=6n - 9=6
  1. n=3n=3
  2. n=15n=15

Exercise 5

3p+6=153p+6=15
  1. p=3p=3

    Answer: yes

  2. p=7p=7

    Answer: no

Exercise 6

8q+4=208q+4=20
  1. q=2q=2
  2. q=3q=3

Exercise 7

18d9=2718d - 9=27
  1. d=1d=1

    Answer: no

  2. d=2d=2

    Answer: yes

Exercise 8

24f12=6024f - 12=60
  1. f=2f=2
  2. f=3f=3

Exercise 9

8u4=4u+408u - 4=4u+40
  1. u=3u=3

    Answer: no

  2. u=11u=11

    Answer: yes

Exercise 10

7v3=4v+367v - 3=4v+36
  1. v=3v=3
  2. v=11v=11

Exercise 11

20h5=15h+3520h - 5=15h+35
  1. h=6h=6

    Answer: no

  2. h=8h=8

    Answer: yes

Exercise 12

18k3=12k+3318k - 3=12k+33
  1. k=1k=1
  2. k=6k=6

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality.

Exercise 1

The image is divided in half vertically. On the left side is an envelope with 2 counters below it. On the right side is 5 counters.

Answer: x + 2 = 5; x = 3

Exercise 2

The image is divided in half vertically. On the left side is an envelope with 4 counters below it. On the right side is 7 counters.

Exercise 3

The image is divided in half vertically. On the left side is an envelope with three counters below it. On the right side is 6 counters.

Answer: x + 3 = 6; x = 3

Exercise 4

The image is divided in half vertically. On the left side is an envelope with 5 counters below it. On the right side is 9 counters.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.
  1. a+2=18a+2=18

    Answer: a = 16

  2. b+5=13b+5=13
  3. p+18=23p+18=23

    Answer: p = 5

  4. q+14=31q+14=31
  5. r+76=100r+76=100

    Answer: r = 24

  6. s+62=95s+62=95
  7. 16=x+916=x+9

    Answer: x = 7

  8. 17=y+617=y+6
  9. 93=p+2493=p+24

    Answer: p = 69

  10. 116=q+79116=q+79
  11. 465=d+398465=d+398

    Answer: d = 67

  12. 932=c+641932=c+641

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.
  1. y3=19y - 3=19

    Answer: y = 22

  2. x4=12x - 4=12
  3. u6=24u - 6=24

    Answer: u = 30

  4. v7=35v - 7=35
  5. f55=123f - 55=123

    Answer: f = 178

  6. g39=117g - 39=117
  7. 19=n1319=n - 13

    Answer: n = 32

  8. 18=m1518=m - 15
  9. 10=p3810=p - 38

    Answer: p = 48

  10. 18=q7218=q - 72
  11. 268=y199268=y - 199

    Answer: y = 467

  12. 204=z149204=z - 149

Translate Word Phrase to Algebraic Equations

In the following exercises, translate the given sentence into an algebraic equation.
  1. The sum of 88 and 99 is equal to 1717.

    Answer: 8 + 9 = 17

  2. The sum of 77 and 99 is equal to 1616.
  3. The difference of 2323 and 1919 is equal to 44.

    Answer: 23 − 19 = 4

  4. The difference of 2929 and 1212 is equal to 1717.
  5. The product of 33 and 99 is equal to 2727.

    Answer: 3 ⋅ 9 = 27

  6. The product of 66 and 88 is equal to 4848.
  7. The quotient of 5454 and 66 is equal to 99.

    Answer: 54 ÷ 6 = 9

  8. The quotient of 4242 and 77 is equal to 66.
  9. Twice the difference of nn and 1010 gives 5252.

    Answer: 2(n − 10) = 52

  10. Twice the difference of mm and 1414 gives 6464.
  11. The sum of three times yy and 1010 is 100100.

    Answer: 3y + 10 = 100

  12. The sum of eight times xx and 44 is 6868.

Translate to an Equation and Solve

In the following exercises, translate the given sentence into an algebraic equation and then solve it.
  1. Five more than pp is equal to 2121.

    Answer: p + 5 = 21; p = 16

  2. Nine more than qq is equal to 4040.
  3. The sum of rr and 1818 is 7373.

    Answer: r + 18 = 73; r = 55

  4. The sum of ss and 1313 is 6868.
  5. The difference of dd and 3030 is equal to 5252.

    Answer: d − 30 = 52; d = 82

  6. The difference of cc and 2525 is equal to 7575.
  7. 1212 less than uu is 8989.

    Answer: u − 12 = 89; u = 101

  8. 1919 less than ww is 5656.
  9. 325325 less than cc gives 799799.

    Answer: c − 325 = 799; c = 1124

  10. 299299 less than dd gives 850850.

Everyday Math

Insurance

Vince’s car insurance has a $500\$500 deductible. Find the amount the insurance company will pay, pp, for an $1800\$1800 claim by solving the equation 500+p=1800500+p=1800.

Answer: $1300

Insurance

Marta’s homeowner’s insurance policy has a $750\$750 deductible. The insurance company paid $5800\$5800 to repair damages caused by a storm. Find the total cost of the storm damage, dd, by solving the equation d750=5800d - 750=5800.

Sale purchase

Arthur bought a suit that was on sale for $120\$120 off. He paid $340\$340 for the suit. Find the original price, pp, of the suit by solving the equation p120=340p - 120=340.

Answer: $460

Sale purchase

Rita bought a sofa that was on sale for $1299\$1299. She paid a total of $1409\$1409, including sales tax. Find the amount of the sales tax, tt, by solving the equation 1299+t=14091299+t=1409.

Writing Exercises

Is x=1x=1 a solution to the equation 8x2=166x?8x - 2=16 - 6x? How do you know? [practice-area rows="4"][/practice-area] Write the equation y5=21y - 5=21 in words. Then make up a word problem for this equation. [practice-area rows="4"][/practice-area]

Finding Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than 5050 for the given number.

  1. 22

    Answer: 2, 4, 6, 8, 10 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48

  2. 33
  3. 44

    Answer: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

  4. 55
  5. 66

    Answer: 6, 12, 18, 24, 30, 36, 42, 48

  6. 77
  7. 88

    Answer: 8, 16, 24, 32, 40, 48

  8. 99
  9. 1010

    Answer: 10, 20, 30, 40

  10. 1212

Use Common Divisibility Tests

In the following exercises, use the divisibility tests to determine whether each number is divisible by 2,3,4,5,6,and102,3,4,5,6,\text{and}10.

  1. 8484

    Answer: Divisible by 2, 3, 4, 6

  2. 9696
  3. 7575

    Answer: Divisible by 3, 5

  4. 7878
  5. 168168

    Answer: Divisible by 2, 3, 4, 6

  6. 264264
  7. 900900

    Answer: Divisible by 2, 3, 4, 5, 6, 10

  8. 800800
  9. 896896

    Answer: Divisible by 2, 4

  10. 942942
  11. 375375

    Answer: Divisible by 3, 5

  12. 750750
  13. 350350

    Answer: Divisible by 2, 5, 10

  14. 550550
  15. 14301430

    Answer: Divisible by 2, 5, 10

  16. 10801080
  17. 22,33522,335

    Answer: Divisible by 3, 5

  18. 39,07539,075

Find All the Factors of a Number

In the following exercises, find all the factors of the given number.

  1. 3636

    Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36

  2. 4242
  3. 6060

    Answer: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

  4. 4848
  5. 144144

    Answer: 1, 2, 3, 4, 6, 8, 12, 18, 24, 36, 48, 72,144

  6. 200200
  7. 588588

    Answer: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588

  8. 576576

Identify Prime and Composite Numbers

In the following exercises, determine if the given number is prime or composite.

  1. 4343

    Answer: prime

  2. 6767
  3. 3939

    Answer: composite

  4. 5353
  5. 7171

    Answer: prime

  6. 119119
  7. 481481

    Answer: composite

  8. 221221
  9. 209209

    Answer: composite

  10. 359359
  11. 667667

    Answer: composite

  12. 17711771

Everyday Math

Banking

Frank’s grandmother gave him $100\$100 at his high school graduation. Instead of spending it, Frank opened a bank account. Every week, he added $15\$15 to the account. The table shows how much money Frank had put in the account by the end of each week. Complete the table by filling in the blanks.
Weeks after graduation Total number of dollars Frank put in the account Simplified Total
00 100100 100100
11 100+15100+15 115115
22 100+152100+15\cdot 2 130130
33 100+153100+15\cdot 3 [practice-area rows="1"][/practice-area]
44 100+15[]100+15\cdot \left[\right] [practice-area rows="1"][/practice-area]
55 100+[]100+\left[\right] [practice-area rows="1"][/practice-area]
66 [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
2020 [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
xx [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]

Answer: This table has nine rows and three columns. The first row is a header row that labels each column. The first column is labeled

Banking

In March, Gina opened a Christmas club savings account at her bank. She deposited $75\$75 to open the account. Every week, she added $20\$20 to the account. The table shows how much money Gina had put in the account by the end of each week. Complete the table by filling in the blanks.
Weeks after opening the account Total number of dollars Gina put in the account Simplified Total
00 7575 7575
11 75+2075+20 9595
22 75+20275+20\cdot 2 115115
33 75+20375+20\cdot 3 [practice-area rows="1"][/practice-area]
44 75+20[]75+20\cdot \left[\right] [practice-area rows="1"][/practice-area]
55 75+[]75+\left[\right] [practice-area rows="1"][/practice-area]
66 [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
2020 [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]
xx [practice-area rows="1"][/practice-area] [practice-area rows="1"][/practice-area]

Writing Exercises

If a number is divisible by 22 and by 33, why is it also divisible by 6?6? [practice-area rows="4"][/practice-area] What is the difference between prime numbers and composite numbers? [practice-area rows="4"][/practice-area]

Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using the factor tree method.

  1. 8686

    Answer: 2 ⋅ 43

  2. 7878
  3. 132132

    Answer: 2 ⋅ 2 ⋅ 3 ⋅ 11

  4. 455455
  5. 693693

    Answer: 3 ⋅ 3 ⋅ 7 ⋅ 11

  6. 420420
  7. 115115

    Answer: 5 ⋅ 23

  8. 225225
  9. 24752475

    Answer: 3 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 11

  10. 1560

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using the ladder method.

  1. 5656

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 7

  2. 7272
  3. 168168

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 7

  4. 252252
  5. 391391

    Answer: 17 ⋅ 23

  6. 400400
  7. 432432

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3

  8. 627627
  9. 21602160

    Answer: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5

  10. 25202520

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number using any method.

  1. 150150

    Answer: 2 ⋅ 3 ⋅ 5 ⋅ 5

  2. 180180
  3. 525525

    Answer: 3 ⋅ 5 ⋅ 5 ⋅ 7

  4. 444444
  5. 3636

    Answer: 2 ⋅ 2 ⋅ 3 ⋅ 3

  6. 5050
  7. 350350

    Answer: 2 ⋅ 5 ⋅ 5 ⋅ 7

  8. 144144

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by listing multiples.
  1. 8,128,12

    Answer: 24

  2. 4,34,3
  3. 6,156,15

    Answer: 30

  4. 12,1612,16
  5. 30,4030,40

    Answer: 120

  6. 20,3020,30
  7. 60,7560,75

    Answer: 300

  8. 44,5544,55

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) by using the prime factors method.

  1. 8,128,12

    Answer: 24

  2. 12,1612,16
  3. 24,3024,30

    Answer: 120

  4. 28,4028,40
  5. 70,8470,84

    Answer: 420

  6. 84,9084,90

Find the Least Common Multiple (LCM) of Two Numbers

In the following exercises, find the least common multiple (LCM) using any method.

  1. 6,216,21

    Answer: 42

  2. 9,159,15
  3. 24,3024,30

    Answer: 120

  4. 32,4032,40

Everyday Math

Grocery shopping

Hot dogs are sold in packages of ten, but hot dog buns come in packs of eight. What is the smallest number of hot dogs and buns that can be purchased if you want to have the same number of hot dogs and buns? (Hint: it is the LCM!)

Answer: 40

Grocery shopping

Paper plates are sold in packages of 1212 and party cups come in packs of 88. What is the smallest number of plates and cups you can purchase if you want to have the same number of each? (Hint: it is the LCM!)

Writing Exercises

Do you prefer to find the prime factorization of a composite number by using the factor tree method or the ladder method? Why? [practice-area rows="4"][/practice-area] Do you prefer to find the LCM by listing multiples or by using the prime factors method? Why? [practice-area rows="4"][/practice-area]

Chapter Review Exercises

Use the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebra to English.
  1. 383\cdot 8

    Answer: the product of 3 and 8

  2. 12x12-x
  3. 24÷624\div 6

    Answer: the quotient of 24 and 6

  4. 9+2a9+2a
  5. 504750\ge 47

    Answer: 50 is greater than or equal to 47

  6. 3y<153y<15
  7. n+4=13n+4=13

    Answer: The sum of n and 4 is equal to 13

  8. 32k=732-k=7

Identify Expressions and Equations

In the following exercises, determine if each is an expression or equation.
  1. 5+u=845+u=84

    Answer: equation

  2. 366s36 - 6s
  3. 4y114y - 11

    Answer: expression

  4. 10x=12010x=120

Simplify Expressions with Exponents

In the following exercises, write in exponential form.
  1. 2222\cdot 2\cdot 2

    Answer: 23

  2. aaaaaa\cdot a\cdot a\cdot a\cdot a
  3. xxxxxxx\cdot x\cdot x\cdot x\cdot x\cdot x

    Answer: x6

  4. 10101010\cdot 10\cdot 10

Simplify Expressions with Exponents

In the following exercises, write in expanded form.
  1. 84{8}^{4}

    Answer: 8 ⋅ 8 ⋅ 8 ⋅ 8

  2. 36{3}^{6}
  3. y5{y}^{5}

    Answer: y ⋅ y ⋅ y ⋅ y ⋅ y

  4. n4{n}^{4}

Simplify Expressions with Exponents

In the following exercises, simplify each expression.
  1. 34{3}^{4}

    Answer: 81

  2. 106{10}^{6}
  3. 27{2}^{7}

    Answer: 128

  4. 43{4}^{3}

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.
  1. 10+2510+2\cdot 5

    Answer: 20

  2. (10+2)5\left(10+2\right)\cdot 5
  3. (30+6)÷2\left(30+6\right)\div 2

    Answer: 18

  4. 30+6÷230+6\div 2
  5. 72+52{7}^{2}+{5}^{2}

    Answer: 74

  6. (7+5)2{\left(7+5\right)}^{2}
  7. 4+3(101)4+3\left(10 - 1\right)

    Answer: 31

  8. (4+3)(101)\left(4+3\right)\left(10 - 1\right)

Evaluate, Simplify, and Translate Expressions

Evaluate an Expression

In the following exercises, evaluate the following expressions.
  1. 9x5 when x=79x - 5\text{ when }x=7

    Answer: 58

  2. y3 when y=5{y}^{3}\text{ when }y=5
  3. 3a4b3a - 4b when a=10,b=1a=10,b=1

    Answer: 26

  4. bh when b=7,h=8bh\text{ when }b=7,h=8

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the terms in each expression.
  1. 12n2+3n+112{n}^{2}+3n+1

    Answer: 12n2,3n, 1

  2. 4x3+11x+34{x}^{3}+11x+3

Identify Terms, Coefficients and Like Terms

In the following exercises, identify the coefficient of each term.
  1. 6y6y

    Answer: 6

  2. 13x213{x}^{2}
In the following exercises, identify the like terms.
  1. 5x2,3,5y2,3x,x,45{x}^{2},3,5{y}^{2},3x,x,4

    Answer: 3, 4, and 3x, x

  2. 8,8r2,8r,3r,r2,3s8,8{r}^{2},\text{8}r,3r,{r}^{2},3s

Simplify Expressions by Combining Like Terms

In the following exercises, simplify the following expressions by combining like terms.
  1. 15a+9a15a+9a

    Answer: 24a

  2. 12y+3y+y12y+3y+y
  3. 4x+7x+3x4x+7x+3x

    Answer: 14x

  4. 6+5c+36+5c+3
  5. 8n+2+4n+98n+2+4n+9

    Answer: 12n + 11

  6. 19p+5+4p1+3p19p+5+4p - 1+3p
  7. 7y2+2y+11+3y287{y}^{2}+2y+11+3{y}^{2}-8

    Answer: 10y2 + 2y + 3

  8. 13x2x+6+5x2+9x13{x}^{2}-x+6+5{x}^{2}+9x

Translate English Phrases to Algebraic Expressions

In the following exercises, translate the following phrases into algebraic expressions.
  1. the difference of xx and 66

    Answer: x − 6

  2. the sum of 1010 and twice aa
  3. the product of 3n3n and 99

    Answer: 3n ⋅ 9

  4. the quotient of ss and 44
  5. 55 times the sum of yy and 11

    Answer: 5(y + 1)

  6. 1010 less than the product of 55 and zz

Translate English Phrases to Algebraic Expressions

In the following exercises, write the algebraic expressions that can be found in each sentence.
  1. Jack bought a sandwich and a coffee. The cost of the sandwich was $3\$3 more than the cost of the coffee. Call the cost of the coffee cc. Write an expression for the cost of the sandwich.

    Answer: c + 3

  2. The number of poetry books on Brianna’s bookshelf is 55 less than twice the number of novels. Call the number of novels nn. Write an expression for the number of poetry books.

Subtraction Property of Equality

Determine Whether a Number is a Solution of an Equation

In the following exercises, determine whether each number is a solution to the equation.
Exercise 1
y+16=40y+16=40
  1. 2424

    Answer: yes

  2. 5656

    Answer: no

Exercise 2
d6=21d - 6=21
  1.  1515
  2. 2727
Exercise 3
4n+12=364n+12=36
  1. 66

    Answer: yes

  2. 1212

    Answer: No

Exercise 4
20q10=7020q - 10=70
  1. 33
  2. 44
Exercise 5
15x5=10x+4515x - 5=10x+45
  1.  22

    Answer: no

  2. 1010

    Answer: yes

Exercise 6
22p6=18p+8622p - 6=18p+86
  1. 44
  2. 2323

Model the Subtraction Property of Equality

In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality. This image is divided into two parts: the first part shows an envelope and 3 blue counters and the next to it, the second part shows five counters.

Answer: x + 3 = 5; x = 2

This image is divided into two parts: the first part shows an envelope and 4 blue counters and next to it, the second part shows 9 counters.

Solve Equations using the Subtraction Property of Equality

In the following exercises, solve each equation using the subtraction property of equality.
  1. c+8=14c+8=14

    Answer: 6

  2. v+8=150v+8=150
  3. 23=x+1223=x+12

    Answer: 11

  4. 376=n+265376=n+265

Solve Equations using the Addition Property of Equality

In the following exercises, solve each equation using the addition property of equality.
  1. y7=16y - 7=16

    Answer: 23

  2. k42=113k - 42=113
  3. 19=p1519=p - 15

    Answer: 34

  4. 501=u399501=u - 399

Translate English Sentences to Algebraic Equations

In the following exercises, translate each English sentence into an algebraic equation.
  1. The sum of 77 and 3333 is equal to 4040.

    Answer: 7 + 33 = 44

  2. The difference of 1515 and 33 is equal to 1212.
  3. The product of 44 and 88 is equal to 3232.

    Answer: 4 ⋅ 8 = 32

  4. The quotient of 6363 and 99 is equal to 77.
  5. Twice the difference of nn and 33 gives 7676.

    Answer: 2(n − 3) = 76

  6. The sum of five times yy and 44 is 8989.

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and then solve it.
  1. Eight more than xx is equal to 3535.

    Answer: x + 8 = 35; x = 27

  2. 2121 less than aa is 1111.
  3. The difference of qq and 1818 is 5757.

    Answer: q − 18 = 57; q = 75

  4. The sum of mm and 125125 is 240240.

Mixed Practice

In the following exercises, solve each equation.
  1. h15=27h - 15=27

    Answer: h = 42

  2. k11=34k - 11=34
  3. z+52=85z+52=85

    Answer: z = 33

  4. x+93=114x+93=114
  5. 27=q+1927=q+19

    Answer: q = 8

  6. 38=p+1938=p+19
  7. 31=v2531=v - 25

    Answer: v = 56

  8. 38=u1638=u - 16

Finding Multiples and Factors

Identify Multiples of Numbers

In the following exercises, list all the multiples less than 5050 for each of the following.
  1. 33

    Answer: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48

  2. 22
  3. 88

    Answer: 8, 16, 24, 32, 40, 48

  4. 1010

Use Common Divisibility Tests

In the following exercises, using the divisibility tests, determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
  1. 9696

    Answer: 2, 3, 6

  2. 250250
  3. 420420

    Answer: 2, 3, 5, 6, 10

  4. 625625

Find All the Factors of a Number

In the following exercises, find all the factors of each number.
  1. 3030

    Answer: 1, 2, 3, 5, 6, 10, 15, 30

  2. 7070
  3. 180180

    Answer: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

  4. 378378

Identify Prime and Composite Numbers

In the following exercises, identify each number as prime or composite.
  1. 1919

    Answer: prime

  2. 5151
  3. 121121

    Answer: composite

  4. 219219

Prime Factorization and the Least Common Multiple

Find the Prime Factorization of a Composite Number

In the following exercises, find the prime factorization of each number.
  1. 8484

    Answer: 2 ⋅ 2 ⋅ 3 ⋅ 7

  2. 165165
  3. 350350

    Answer: 2 ⋅ 5 ⋅ 5 ⋅ 7

  4. 572572

Find the Least Common Multiple of Two Numbers

In the following exercises, find the least common multiple of each pair of numbers.
  1. 9,159,15

    Answer: 45

  2. 12,2012,20
  3. 25,3525,35

    Answer: 350

  4. 18,4018,40

Everyday Math

Describe how you have used two topics from The Language of Algebra chapter in your life outside of your math class during the past month. [practice-area rows="4"][/practice-area]

Chapter Practice Test

In the following exercises, translate from an algebraic equation to English phrases.
  1. 646\cdot 4
  2. 15x15-x

    Answer: fifteen minus x

In the following exercises, identify each as an expression or equation.
  1. 58+105\cdot 8+10
  2. x+6=9x+6=9

    Answer: equation

  3. 311=333\cdot 11=33
  4. Write nnnnnnn\cdot n\cdot n\cdot n\cdot n\cdot n in exponential form.

    Answer: n6

  5. Write 35{3}^{5} in expanded form and then simplify.

    Answer: 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 243

In the following exercises, simplify, using the order of operations.
  1. 4+354+3\cdot 5
  2. (8+1)4\left(8+1\right)\cdot 4

    Answer: 36

  3. 1+6(31)1+6\left(3 - 1\right)
  4. (8+4)÷3+1\left(8+4\right)\div 3+1

    Answer: 5

  5. (1+4)2{\left(1+4\right)}^{2}
  6. 5[2+7(98)]5\left[2+7\left(9 - 8\right)\right]

    Answer: 45

In the following exercises, evaluate each expression.
  1. 8x3 when x=48x - 3\text{ when }x=4
  2. y3 when y=5{y}^{3}\text{ when }y=5

    Answer: 125

  3. 6a2b when a=5,b=76a - 2b\text{ when }a=5,b=7
  4. hw when h=12,w=3hw\text{ when }h=12,w=3

    Answer: 36

Simplify by combining like terms.
  1.  6x+8x6x+8x
  2. 9m+10+m+39m+10+m+3
In the following exercises, translate each phrase into an algebraic expression.
  1. 55 more than xx

    Answer: x + 5

  2. the quotient of 1212 and yy
  3. three times the difference of a and ba\text{ and }b

    Answer: 3(a − b)

  4. Caroline has 33 fewer earrings on her left ear than on her right ear. Call the number of earrings on her right ear, rr. Write an expression for the number of earrings on her left ear.
In the following exercises, solve each equation.
  1. n6=25n - 6=25

    Answer: n = 31

  2. x+58=71x+58=71
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
  1. 1515 less than yy is 3232.

    Answer: y − 15 = 32; y = 47

  2. the sum of aa and 129129 is 164164.
  3. List all the multiples of 44, that are less than 5050.

    Answer: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48

  4. Find all the factors of 9090.
  5. Find the prime factorization of 10801080.

    Answer: 23 ⋅ 33 ⋅ 5

  6. Find the LCM (Least Common Multiple) of 2424 and 4040.

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