The People’s Share

GED Math

Algebra Practice 1

Variables, expressions, and equations. More than three hundred problems, from the first idea to solving equations. The answer key is at the end.

← Math · GED at The People’s Share

This is a practice sheet. It starts with the first idea in algebra, that a letter can stand for a number, and works up to solving equations. Each part begins with the easiest version and adds one thing at a time. The problems are numbered from 1 to the end, and the answer key uses the same numbers.

Work in order. Do a group of problems, check them against the key, and then go on. If a section is easy, do a few and move ahead. If a section is hard, stay with it. There are more problems here than anyone needs in one sitting, so come back to the sheet more than once. Write your answers on paper, or print the sheet and write on it.

This sheet goes with the algebra pages under the apps on the Math door: The Thing Called x, The Balance, and The Share. The pages explain the ideas with pictures; this sheet is for practice.

How things are written in algebra

Part One

The variable stands for a number

1.One variable, one number

In each problem the variable has been given a number. Replace the variable with its number, then do the arithmetic.

Examplek = 6. What is k + 1? Replace k with 6: 6 + 1 = 7.

Find the value.

  1. 1.m = 2. m + m =
  2. 2.m = 2. m + 3 =
  3. 3.m = 2. m + 10 =
  4. 4.m = 2. 5 + m =
  5. 5.x = 5. x + 3 =
  6. 6.x = 5. x + x =
  7. 7.x = 5. x − 1 =
  8. 8.x = 5. 12 − x =
  9. 9.y = 8. y − 8 =
  10. 10.y = 8. y + y =
  11. 11.y = 8. 20 − y =
  12. 12.n = 4. n × 3 =
  13. 13.n = 4. n × n =
  14. 14.n = 4. n ÷ 2 =
  15. 15.n = 4. 2 × n =
  16. 16.a = 10. a ÷ 5 =
  17. 17.a = 10. a − 4 =
  18. 18.a = 10. a + a =
  19. 19.k = 0. k + 7 =
  20. 20.k = 0. k × 9 =
  21. 21.p = 1. p + p =
  22. 22.p = 1. p × 6 =
  23. 23.w = 12. w ÷ 4 =
  24. 24.w = 12. w − w =

2.The same variable, added again and again

x + x is two xs. In algebra that is written 2x, and it means 2 times x. z + z + z is three zs, written 3z. So when you see a number written right next to a variable, multiply.

Examplea = 4. What is a + a + a? The long way: 4 + 4 + 4 = 12. The short way: a + a + a is 3a, and 3a = 3 × 4 = 12. Same answer.

Find the value. If it helps, count the variables first.

  1. 25.x = 7. x + x =
  2. 26.x = 7. 2x =
  3. 27.z = 3. z + z + z =
  4. 28.z = 3. 3z =
  5. 29.m = 5. m + m + m + m =
  6. 30.m = 5. 4m =
  7. 31.y = 2. 5y =
  8. 32.y = 2. y + y + y + y + y =
  9. 33.b = 6. 2b =
  10. 34.b = 6. 3b =
  11. 35.b = 6. 10b =
  12. 36.n = 9. 2n =
  13. 37.n = 9. n + n + n =
  14. 38.k = 0. 5k =
  15. 39.k = 1. 5k =
  16. 40.t = 10. 3t =
  17. 41.t = 10. 7t =
  18. 42.c = 8. 2c =
  19. 43.c = 8. c + c + c =
  20. 44.h = 12. 2h =
  21. 45.h = 12. 5h =
  22. 46.h = 12. h + h + h + h + h + h =

3.Which means the same?

Each problem gives one expression and three choices. Only one choice means the same thing.

Examplez + z + z + z means the same as: A) 4z  B) z + 4  C) z². The answer is A, because four zs added together is 4 times z.

Choose the letter of the right answer.

  1. 47.x + x: A) x²  B) 2x  C) x + 2
  2. 48.y + y + y: A) 3y  B) y + 3  C) y³
  3. 49.4m: A) m + 4  B) m ÷ 4  C) 4 × m
  4. 50.2a + 3: A) 2 × (a + 3)  B) 2 × a, then add 3  C) 2 + a + 3
  5. 51.n × n: A) 2n  B) n + n  C) n²
  6. 52.b ÷ 3: A) b/3  B) 3b  C) 3 − b
  7. 53.5p − 1: A) 5 × (p − 1)  B) 5 + p − 1  C) 5 × p, then subtract 1
  8. 54.k + k + 6: A) 2k + 6  B) k + 6  C) 8k

4.Two variables

Now there are two variables, each with its own number. Replace each variable with its own number. Remember that xy means x times y. Two different variables can hold the same number.

Examplec = 2, d = 9. Then c + d = 2 + 9 = 11. And cd = 2 × 9 = 18. And 3c + d = 6 + 9 = 15.

Find the value.

  1. 55.x = 3, y = 5. x + y =
  2. 56.x = 3, y = 5. y − x =
  3. 57.x = 3, y = 5. x × y =
  4. 58.x = 3, y = 5. xy =
  5. 59.x = 3, y = 5. 2x + y =
  6. 60.x = 3, y = 5. x + 2y =
  7. 61.x = 3, y = 5. 2x + 2y =
  8. 62.x = 3, y = 5. 3x − y =
  9. 63.a = 10, b = 2. a ÷ b =
  10. 64.a = 10, b = 2. a − b =
  11. 65.a = 10, b = 2. ab =
  12. 66.a = 10, b = 2. 3a + b =
  13. 67.a = 10, b = 2. a + b + b =
  14. 68.a = 10, b = 2. a ÷ b + 1 =
  15. 69.m = 4, n = 6. m + n =
  16. 70.m = 4, n = 6. n − m =
  17. 71.m = 4, n = 6. mn =
  18. 72.m = 4, n = 6. 2m + n =
  19. 73.m = 4, n = 6. m + 2n =
  20. 74.m = 4, n = 6. 5m − 2n =
  21. 75.p = 7, q = 7. p + q =
  22. 76.p = 7, q = 7. p − q =
  23. 77.p = 7, q = 7. pq =
  24. 78.r = 0, s = 9. rs =
  25. 79.r = 0, s = 9. r + s =
  26. 80.r = 0, s = 9. s − r =

Part Two

From words to symbols

An expression is a piece of math with no equals sign: n + 4, 2x, 3y − 10, (n + 4) ÷ 6. In this part you turn words into expressions, and then expressions back into words. On the GED test, and in life, the math usually arrives as words first.

5.Plus, minus, times, divided by

Write each phrase in symbols. Use n for “a number.” If a phrase has two different unknown numbers, use n for the first and m for the second. Watch the order: “a number minus 8” is n − 8, but “8 minus a number” is 8 − n.

Examplea number minus 3 → n − 3.    a number multiplied by 5 → 5n, which is the same as n × 5.

Write the expression.

  1. 81.a number plus 4
  2. 82.a number plus 9
  3. 83.4 plus a number
  4. 84.a number minus 8
  5. 85.a number minus 1
  6. 86.8 minus a number
  7. 87.a number times 3
  8. 88.a number multiplied by 10
  9. 89.6 times a number
  10. 90.a number divided by 2
  11. 91.a number divided by 5
  12. 92.20 divided by a number
  13. 93.a number multiplied by itself
  14. 94.a number added to itself
  15. 95.a number plus another number
  16. 96.a number multiplied by another number
  17. 97.a number minus another number
  18. 98.a number divided by another number
  19. 99.1 times a number
  20. 100.a number plus 0

6.Other words for the same thing

Many words point to the same four operations.
Add: sum, plus, more than, increased by, added to.
Subtract: difference, minus, less than, decreased by, take away.
Multiply: product, times, multiplied by; twice and double mean times 2; triple means times 3.
Divide: quotient, divided by; half of means divided by 2; a third of means divided by 3.
One trap: “6 less than a number” means take 6 away from the number, so it is n − 6. The number comes first even though the phrase says 6 first.

Example5 more than a number → n + 5.    5 less than a number → n − 5.    the product of 4 and a number → 4n.

Write the expression. Use n for the number, and n and m for two different numbers.

  1. 101.the sum of a number and 7
  2. 102.the sum of 12 and a number
  3. 103.the difference between a number and 4
  4. 104.the product of a number and 9
  5. 105.the product of 2 and a number
  6. 106.the quotient of a number and 3
  7. 107.twice a number
  8. 108.double a number
  9. 109.half of a number
  10. 110.a third of a number
  11. 111.6 more than a number
  12. 112.6 less than a number
  13. 113.a number increased by 15
  14. 114.a number decreased by 15
  15. 115.a number squared
  16. 116.three times a number
  17. 117.a number tripled
  18. 118.the sum of two numbers
  19. 119.the product of two numbers
  20. 120.10 more than a number
  21. 121.10 less than a number
  22. 122.10 decreased by a number
  23. 123.the quotient of 100 and a number
  24. 124.the sum of a number and itself

7.Two steps in one phrase

Some phrases have two operations, and the order matters. A comma often shows it. “Twice a number, plus 3” means: double the number, then add 3. That is 2n + 3. “Twice the sum of a number and 3” means: add 3 first, then double. That is 2(n + 3). The parentheses say “do this part first.”

Examplea number minus 5, divided by 2 → (n − 5) ÷ 2.    1 more than four times a number → 4n + 1.

Write the expression.

  1. 125.twice a number, plus 5
  2. 126.twice a number, minus 5
  3. 127.5 more than twice a number
  4. 128.three times a number, plus 1
  5. 129.1 less than three times a number
  6. 130.a number plus 4, divided by 6
  7. 131.a number minus 2, divided by 3
  8. 132.a number divided by 2, plus 7
  9. 133.half of a number, plus 7
  10. 134.twice the sum of a number and 7
  11. 135.4 times the difference between a number and 1
  12. 136.the sum of twice a number and 9
  13. 137.10 minus twice a number
  14. 138.a number squared, plus 1
  15. 139.the product of a number and 5, minus 2
  16. 140.100 divided by a number, plus 1
  17. 141.the sum of a number and another number, times 2
  18. 142.the sum of a number and 8, divided by 2

8.From symbols to words

Now go the other way. Write each expression in words. There is more than one right way to say each one; the key shows one or two.

Examplen − 2 → a number minus 2, or 2 less than a number.

Write the words.

  1. 143.n + 6
  2. 144.n − 9
  3. 145.9 − n
  4. 146.4n
  5. 147.n ÷ 8
  6. 148.2n + 1
  7. 149.n²
  8. 150.(n + 5) ÷ 2
  9. 151.3n − 4
  10. 152.n + m
  11. 153.5(n − 1)
  12. 154.12 ÷ n

Part Three

Writing expressions more simply

An expression can often be written in a shorter way that means the same thing. x + x + x is 3x. 2x + 3x is 5x, because two xs and three xs make five xs. Writing an expression in its shortest form is called simplifying.

9.Collect what is alike

Each piece of an expression that is added or subtracted, like 2x, 3y, or 5, is called a term. Add or subtract the terms that have the same variable. Terms with different variables stay separate: 2x + 3y cannot be made shorter. A plain number with no variable stays by itself, but plain numbers can be combined with each other.

Example4a + a = 5a.    7k − 2k = 5k.    3x + 2 + x = 4x + 2.    2x + 3y stays 2x + 3y.

Write each expression in its shortest form. If it is already as short as it can be, write it again as it is.

  1. 155.m + m
  2. 156.a + a + a + a
  3. 157.2x + 3x
  4. 158.5y + y
  5. 159.7k − 4k
  6. 160.10n − n
  7. 161.3b + 3b
  8. 162.8p − 8p
  9. 163.x + x + y + y
  10. 164.3x + 2y + x
  11. 165.4a + 5 + a
  12. 166.2n + 3 + 3
  13. 167.6t − 2t + 1
  14. 168.5m + 2 − 2
  15. 169.x + 3x + 5x
  16. 170.9c − 3c − c
  17. 171.2x + 3y
  18. 172.7 + 2h + 3
  19. 173.a + b + a + b + a
  20. 174.12k − 5k + 2k
  21. 175.y + y + y − y
  22. 176.4x + 1 + 4x + 1

10.A number times a term

2(3z) means 2 times 3z. Three zs, taken two times, is six zs: 2(3z) = 6z. Multiply the numbers; the variable stays.

Example3(4y) = 12y.    5(2m) = 10m.    2(x) = 2x.

Write each expression in its shortest form.

  1. 177.2(3z)
  2. 178.3(2y)
  3. 179.4(5m)
  4. 180.5(2a)
  5. 181.10(3x)
  6. 182.2(2k)
  7. 183.3(3n)
  8. 184.7(x)
  9. 185.2(x)
  10. 186.1(9b)
  11. 187.0(4c)
  12. 188.2(3z) + z
  13. 189.3(2y) − y
  14. 190.4(2m) + 2m
  15. 191.2(5x) − 3x
  16. 192.3(2p) + 2(3p)

11.Simplify, then find the value

Two steps. First write the expression in its shortest form. Then replace the variable with its number. Write both: the short form and the value.

Examplez = 4. z + z + z + z = ? Short form: 4z. Value: 4 × 4 = 16. Check the long way: 4 + 4 + 4 + 4 = 16.

Write the short form and the value.

  1. 193.z = 4. z + z + z =
  2. 194.z = 4. 2(3z) =
  3. 195.x = 7. 2x + 3x =
  4. 196.x = 7. 6x − x =
  5. 197.m = 3. 4(2m) =
  6. 198.m = 3. m + m + 10 =
  7. 199.y = 5. 3y + y + 2 =
  8. 200.y = 5. 10y − 4y =
  9. 201.a = 2, b = 6. a + a + b =
  10. 202.a = 2, b = 6. 3a + 2b + a =
  11. 203.k = 9. 5(2k) =
  12. 204.k = 9. k + 2k + 3k =
  13. 205.n = 1. 7n + 8n =
  14. 206.n = 0. 7n + 8n =
  15. 207.h = 12. 2(3h) − 6h =
  16. 208.h = 12. 4h + 1 − 3h =

Part Four

Longer expressions with a value

12.Replace the variable, then follow the order of operations

Replace each variable with its number. Then do what is inside parentheses first, then multiplying and dividing, then adding and subtracting.

Examplex = 4. 3x + 2 = 3 × 4 + 2 = 12 + 2 = 14.    x = 4. 3(x + 2) = 3 × (4 + 2) = 3 × 6 = 18. The parentheses changed the answer.

Find the value.

  1. 209.x = 5. 2x + 3 =
  2. 210.x = 5. 3x − 5 =
  3. 211.x = 5. 4x + 1 =
  4. 212.x = 5. 20 − 2x =
  5. 213.y = 7. 3y − 10 =
  6. 214.y = 7. 2y + 6 =
  7. 215.y = 7. 5y − 5 =
  8. 216.y = 7. 50 − 7y =
  9. 217.n = 8. (n + 4) ÷ 6 =
  10. 218.n = 8. (n − 2) ÷ 3 =
  11. 219.n = 8. n ÷ 2 + 7 =
  12. 220.n = 8. n ÷ 4 + n ÷ 2 =
  13. 221.m = 3. 2(m + 3) =
  14. 222.m = 3. 2m + 3 =
  15. 223.m = 3. 4(m − 1) =
  16. 224.m = 3. m × m =
  17. 225.m = 3. m² + 1 =
  18. 226.a = 10. a² =
  19. 227.a = 10. a² − a =
  20. 228.a = 10. (a + 2) ÷ 4 =
  21. 229.a = 10. 3a + 2a =
  22. 230.k = 6. 100 − 10k =
  23. 231.k = 6. 2k + 2k + 2 =
  24. 232.k = 6. k(k + 1) =
  25. 233.x = 2, y = 3. 2x + 3y =
  26. 234.x = 2, y = 3. xy + 1 =
  27. 235.x = 2, y = 3. 3x − 2y =
  28. 236.x = 2, y = 3. 2(x + y) =
  29. 237.x = 2, y = 3. x² + y² =
  30. 238.p = 12. p/3 + p/4 =

Part Five

Solving equations

An equation has an equals sign. It says that two things are the same. x + 2 = 9 says: some number, plus 2, is 9. To solve an equation is to find the number that makes it true. Here x = 7, because 7 + 2 = 9. The number that works is called the solution.

Three ideas do all the work.
1. To solve, get the variable by itself on one side of the equals sign.
2. To get it by itself, undo what was done to it. Undo adding by subtracting. Undo subtracting by adding. Undo multiplying by dividing. Undo dividing by multiplying.
3. Whatever you do to one side, do to the other side too. Then the two sides stay equal.

Always check: put your answer back into the equation and see whether it comes out true.

13.Adding and subtracting

x + 2 = 9. The 2 was added to x, so subtract 2 from both sides: x + 2 − 2 = 9 − 2, which is x = 7. Check: 7 + 2 = 9. True.

y − 5 = 8. The 5 was subtracted from y, so add 5 to both sides: y = 13. Check: 13 − 5 = 8. True.

Examplen − 6 = 6. Add 6 to both sides: n = 12. Check: 12 − 6 = 6. True.

Solve each equation, then check.

  1. 239.x + 3 = 10    x =
  2. 240.x + 5 = 12    x =
  3. 241.x + 1 = 9    x =
  4. 242.x + 10 = 25    x =
  5. 243.4 + x = 11    x =
  6. 244.9 + x = 9    x =
  7. 245.y − 4 = 6    y =
  8. 246.y − 2 = 13    y =
  9. 247.y − 10 = 0    y =
  10. 248.y − 7 = 7    y =
  11. 249.n + 8 = 8    n =
  12. 250.n − 8 = 8    n =
  13. 251.n + 20 = 50    n =
  14. 252.n − 25 = 25    n =
  15. 253.15 = m + 6    m =
  16. 254.20 = m − 3    m =
  17. 255.a + 2.5 = 6    a =
  18. 256.a − 1.5 = 4    a =
  19. 257.k + 100 = 101    k =
  20. 258.k − 99 = 1    k =
  21. 259.12 = 5 + t    t =
  22. 260.0 = t − 6    t =
  23. 261.b + 3 = 3    b =
  24. 262.b − 3 = 3    b =

14.Sharing and multiplying

2x = 14 says: two xs together make 14. Share 14 into two equal parts, 7 and 7. So x = 7. In symbols, divide both sides by 2: x = 14 ÷ 2 = 7. Check: 2 × 7 = 14. True.

x ÷ 3 = 4 says: a number shared into three parts gives 4 each. Multiply both sides by 3: x = 12. Check: 12 ÷ 3 = 4. True.

Sometimes the sharing does not come out even. 2k = 7 says two ks make 7. Share 7 into two equal parts: 3 and a half each. So k = 3½, which can also be written 7/2 or 3.5. Check: 2 × 3.5 = 7. True.

Example5m = 35. Divide both sides by 5: m = 7. Check: 5 × 7 = 35. True.

Solve, then check. y/4 means y ÷ 4. When the answer is not a whole number, a fraction, a mixed number, or a decimal is fine; any of the three is right.

  1. 263.2x = 12    x =
  2. 264.2x = 20    x =
  3. 265.3x = 12    x =
  4. 266.3x = 30    x =
  5. 267.5y = 25    y =
  6. 268.4y = 24    y =
  7. 269.10y = 70    y =
  8. 270.6n = 6    n =
  9. 271.7n = 0    n =
  10. 272.8n = 64    n =
  11. 273.12 = 3m    m =
  12. 274.45 = 9m    m =
  13. 275.x ÷ 3 = 5    x =
  14. 276.x ÷ 2 = 9    x =
  15. 277.y/4 = 3    y =
  16. 278.y/10 = 10    y =
  17. 279.n/5 = 0    n =
  18. 280.n/6 = 6    n =
  19. 281.2k = 5    k =
  20. 282.2k = 9    k =
  21. 283.4k = 10    k =
  22. 284.3k = 7    k =
  23. 285.5k = 2    k =
  24. 286.4k = 2    k =
  25. 287.100 = 4t    t =
  26. 288.t/8 = 1.5    t =
  27. 289.9 = 2w    w =
  28. 290.w/3 = 2.5    w =

15.Two steps

When a variable has been multiplied and then had something added or subtracted, undo it in two steps: first the adding or subtracting, then the multiplying or dividing.

4y − 3 = 9. Add 3 to both sides: 4y = 12. Divide both sides by 4: y = 3. Check: 4 × 3 − 3 = 12 − 3 = 9. True.

x/3 + 2 = 5. Subtract 2 from both sides: x/3 = 3. Multiply both sides by 3: x = 9. Check: 9 ÷ 3 + 2 = 3 + 2 = 5. True.

Example5m + 4 = 19. Subtract 4 from both sides: 5m = 15. Divide both sides by 5: m = 3. Check: 5 × 3 + 4 = 19. True.

Solve, then check.

  1. 291.2x + 3 = 11    x =
  2. 292.2x + 1 = 15    x =
  3. 293.2x − 3 = 11    x =
  4. 294.2x − 5 = 5    x =
  5. 295.3y − 10 = 11    y =
  6. 296.3y + 10 = 31    y =
  7. 297.3y − 1 = 20    y =
  8. 298.3y + 3 = 3    y =
  9. 299.5n + 1 = 26    n =
  10. 300.5n − 5 = 20    n =
  11. 301.4m − 3 = 13    m =
  12. 302.4m + 4 = 40    m =
  13. 303.10k + 5 = 55    k =
  14. 304.10k − 10 = 90    k =
  15. 305.x/2 + 1 = 6    x =
  16. 306.x/2 − 1 = 6    x =
  17. 307.x/3 + 4 = 6    x =
  18. 308.x/5 − 2 = 0    x =
  19. 309.6a + 2 = 20    a =
  20. 310.6a − 2 = 22    a =
  21. 311.20 = 4t + 4    t =
  22. 312.1 = 2t − 9    t =
  23. 313.7 + 2b = 19    b =
  24. 314.100 − 2b = 80    b =
  25. 315.2c + 3 = 8    c =
  26. 316.4c − 1 = 9    c =
  27. 317.12 = 3d − 3    d =
  28. 318.9 = 5 + 2d    d =

16.From words to an equation to the answer

Write the equation first, then solve it. Use n for the number. Write both the equation and the answer.

ExampleA number plus 5 is 12. Equation: n + 5 = 12. Solve: n = 7. Check: 7 + 5 = 12. True.

Write the equation and solve it.

  1. 319.A number plus 6 is 15.
  2. 320.A number minus 7 is 9.
  3. 321.Twice a number is 18.
  4. 322.Three times a number is 27.
  5. 323.Half of a number is 6.
  6. 324.A number divided by 4 is 5.
  7. 325.4 more than a number is 11.
  8. 326.4 less than a number is 11.
  9. 327.Twice a number, plus 3, is 11.
  10. 328.Three times a number, minus 1, is 14.
  11. 329.5 more than twice a number is 25.
  12. 330.Half of a number, plus 1, is 7.
  13. 331.A number added to itself is 30.
  14. 332.10 minus a number is 4.
  15. 333.The sum of a number and 9 is 9.
  16. 334.The sum of twice a number and 8 is 20.

17.Is it a solution?

A number is a solution if it makes the equation true. Put the number in and see. Answer yes or no. If the answer is no, find the number that does work.

ExampleIs x = 4 a solution of 3x + 2 = 14? Put in 4: 3 × 4 + 2 = 12 + 2 = 14. Yes.

Answer yes or no. If no, give the solution.

  1. 335.x + 5 = 9.  Is x = 4 a solution?
  2. 336.x − 3 = 8.  Is x = 5 a solution?
  3. 337.2x = 14.  Is x = 7 a solution?
  4. 338.3x = 12.  Is x = 9 a solution?
  5. 339.x/2 = 6.  Is x = 3 a solution?
  6. 340.2x + 1 = 9.  Is x = 4 a solution?
  7. 341.3x − 2 = 10.  Is x = 3 a solution?
  8. 342.5x + 5 = 30.  Is x = 5 a solution?
  9. 343.x + x = 10.  Is x = 5 a solution?
  10. 344.4x − 4 = 0.  Is x = 1 a solution?
  11. 345.10 − x = 3.  Is x = 13 a solution?
  12. 346.x/4 + 1 = 4.  Is x = 12 a solution?

18.A little further

These use everything above, and each adds one small new thing. Some equations have the variable on both sides: 5x = 3x + 8. Subtract 3x from both sides to bring the xs together: 2x = 8, so x = 4. Some have parentheses: 2(x + 1) = 10. Divide both sides by 2 first: x + 1 = 5, so x = 4. Some have two terms with the same variable on one side: collect them first.

Example4x = x + 9. Subtract x from both sides: 3x = 9. So x = 3. Check: 4 × 3 = 12, and 3 + 9 = 12. True.

Solve, then check.

  1. 347.2(x + 3) = 14    x =
  2. 348.3(y − 1) = 12    y =
  3. 349.4(n + 2) = 20    n =
  4. 350.2x + 3x = 25    x =
  5. 351.4y − y = 21    y =
  6. 352.x + x + 2 = 12    x =
  7. 353.5x = 3x + 8    x =
  8. 354.7n = 2n + 25    n =
  9. 355.3m + 4 = m + 10    m =
  10. 356.2k + 1 = k + 6    k =
  11. 357.(x + 2) ÷ 3 = 4    x =
  12. 358.(2y − 1) ÷ 3 = 3    y =

Answer Key

Check your answers here. If one is wrong, work it again before going on. For the word problems, there is often more than one right way to write the answer; the key shows one or two.

1.One variable, one number (1–24)

  1. 1.4
  2. 2.5
  3. 3.12
  4. 4.7
  5. 5.8
  6. 6.10
  7. 7.4
  8. 8.7
  9. 9.0
  10. 10.16
  11. 11.12
  12. 12.12
  13. 13.16
  14. 14.2
  15. 15.8
  16. 16.2
  17. 17.6
  18. 18.20
  19. 19.7
  20. 20.0
  21. 21.2
  22. 22.6
  23. 23.3
  24. 24.0

2.The same variable, added again and again (25–46)

  1. 25.14
  2. 26.14
  3. 27.9
  4. 28.9
  5. 29.20
  6. 30.20
  7. 31.10
  8. 32.10
  9. 33.12
  10. 34.18
  11. 35.60
  12. 36.18
  13. 37.27
  14. 38.0
  15. 39.5
  16. 40.30
  17. 41.70
  18. 42.16
  19. 43.24
  20. 44.24
  21. 45.60
  22. 46.72

3.Which means the same? (47–54)

  1. 47.B) 2x
  2. 48.A) 3y
  3. 49.C) 4 × m
  4. 50.B) 2 × a, then add 3
  5. 51.C) n²
  6. 52.A) b/3
  7. 53.C) 5 × p, then subtract 1
  8. 54.A) 2k + 6

4.Two variables (55–80)

  1. 55.8
  2. 56.2
  3. 57.15
  4. 58.15
  5. 59.11
  6. 60.13
  7. 61.16
  8. 62.4
  9. 63.5
  10. 64.8
  11. 65.20
  12. 66.32
  13. 67.14
  14. 68.6
  15. 69.10
  16. 70.2
  17. 71.24
  18. 72.14
  19. 73.16
  20. 74.8
  21. 75.14
  22. 76.0
  23. 77.49
  24. 78.0
  25. 79.9
  26. 80.9

5.Plus, minus, times, divided by (81–100)

  1. 81.n + 4
  2. 82.n + 9
  3. 83.4 + n
  4. 84.n − 8
  5. 85.n − 1
  6. 86.8 − n
  7. 87.3n (or n × 3)
  8. 88.10n
  9. 89.6n
  10. 90.n ÷ 2 (or n/2)
  11. 91.n ÷ 5 (or n/5)
  12. 92.20 ÷ n (or 20/n)
  13. 93.n × n (or n²)
  14. 94.n + n (or 2n)
  15. 95.n + m
  16. 96.n × m (or nm)
  17. 97.n − m
  18. 98.n ÷ m (or n/m)
  19. 99.1n, which is just n
  20. 100.n + 0, which is just n

6.Other words for the same thing (101–124)

  1. 101.n + 7
  2. 102.12 + n
  3. 103.n − 4
  4. 104.9n
  5. 105.2n
  6. 106.n ÷ 3 (or n/3)
  7. 107.2n
  8. 108.2n
  9. 109.n ÷ 2 (or n/2)
  10. 110.n ÷ 3 (or n/3)
  11. 111.n + 6
  12. 112.n − 6
  13. 113.n + 15
  14. 114.n − 15
  15. 115.n² (or n × n)
  16. 116.3n
  17. 117.3n
  18. 118.n + m
  19. 119.nm (or n × m)
  20. 120.n + 10
  21. 121.n − 10
  22. 122.10 − n
  23. 123.100 ÷ n (or 100/n)
  24. 124.n + n (or 2n)

7.Two steps in one phrase (125–142)

  1. 125.2n + 5
  2. 126.2n − 5
  3. 127.2n + 5
  4. 128.3n + 1
  5. 129.3n − 1
  6. 130.(n + 4) ÷ 6
  7. 131.(n − 2) ÷ 3
  8. 132.n ÷ 2 + 7 (or n/2 + 7)
  9. 133.n/2 + 7
  10. 134.2(n + 7)
  11. 135.4(n − 1)
  12. 136.2n + 9
  13. 137.10 − 2n
  14. 138.n² + 1
  15. 139.5n − 2
  16. 140.100 ÷ n + 1 (or 100/n + 1)
  17. 141.2(n + m)
  18. 142.(n + 8) ÷ 2

8.From symbols to words (143–154)

  1. 143.a number plus 6; or 6 more than a number; or the sum of a number and 6
  2. 144.a number minus 9; or 9 less than a number
  3. 145.9 minus a number
  4. 146.4 times a number; or the product of 4 and a number
  5. 147.a number divided by 8
  6. 148.twice a number, plus 1; or 1 more than twice a number
  7. 149.a number squared; or a number multiplied by itself
  8. 150.a number plus 5, divided by 2; or half of the sum of a number and 5
  9. 151.three times a number, minus 4; or 4 less than three times a number
  10. 152.the sum of two numbers; or a number plus another number
  11. 153.5 times the difference between a number and 1; or a number minus 1, times 5
  12. 154.12 divided by a number; or the quotient of 12 and a number

9.Collect what is alike (155–176)

  1. 155.2m
  2. 156.4a
  3. 157.5x
  4. 158.6y
  5. 159.3k
  6. 160.9n
  7. 161.6b
  8. 162.0
  9. 163.2x + 2y
  10. 164.4x + 2y
  11. 165.5a + 5
  12. 166.2n + 6
  13. 167.4t + 1
  14. 168.5m
  15. 169.9x
  16. 170.5c
  17. 171.2x + 3y (already as short as it can be)
  18. 172.2h + 10
  19. 173.3a + 2b
  20. 174.9k
  21. 175.2y
  22. 176.8x + 2

10.A number times a term (177–192)

  1. 177.6z
  2. 178.6y
  3. 179.20m
  4. 180.10a
  5. 181.30x
  6. 182.4k
  7. 183.9n
  8. 184.7x
  9. 185.2x
  10. 186.9b
  11. 187.0
  12. 188.7z
  13. 189.5y
  14. 190.10m
  15. 191.7x
  16. 192.12p

11.Simplify, then find the value (193–208)

  1. 193.3z = 12
  2. 194.6z = 24
  3. 195.5x = 35
  4. 196.5x = 35
  5. 197.8m = 24
  6. 198.2m + 10 = 16
  7. 199.4y + 2 = 22
  8. 200.6y = 30
  9. 201.2a + b = 10
  10. 202.4a + 2b = 20
  11. 203.10k = 90
  12. 204.6k = 54
  13. 205.15n = 15
  14. 206.15n = 0
  15. 207.0 = 0
  16. 208.h + 1 = 13

12.Replace the variable, then follow the order of operations (209–238)

  1. 209.13
  2. 210.10
  3. 211.21
  4. 212.10
  5. 213.11
  6. 214.20
  7. 215.30
  8. 216.1
  9. 217.2
  10. 218.2
  11. 219.11
  12. 220.6
  13. 221.12
  14. 222.9
  15. 223.8
  16. 224.9
  17. 225.10
  18. 226.100
  19. 227.90
  20. 228.3
  21. 229.50
  22. 230.40
  23. 231.26
  24. 232.42
  25. 233.13
  26. 234.7
  27. 235.0
  28. 236.10
  29. 237.13
  30. 238.7

13.Adding and subtracting (239–262)

  1. 239.x = 7
  2. 240.x = 7
  3. 241.x = 8
  4. 242.x = 15
  5. 243.x = 7
  6. 244.x = 0
  7. 245.y = 10
  8. 246.y = 15
  9. 247.y = 10
  10. 248.y = 14
  11. 249.n = 0
  12. 250.n = 16
  13. 251.n = 30
  14. 252.n = 50
  15. 253.m = 9
  16. 254.m = 23
  17. 255.a = 3.5
  18. 256.a = 5.5
  19. 257.k = 1
  20. 258.k = 100
  21. 259.t = 7
  22. 260.t = 6
  23. 261.b = 0
  24. 262.b = 6

14.Sharing and multiplying (263–290)

  1. 263.x = 6
  2. 264.x = 10
  3. 265.x = 4
  4. 266.x = 10
  5. 267.y = 5
  6. 268.y = 6
  7. 269.y = 7
  8. 270.n = 1
  9. 271.n = 0
  10. 272.n = 8
  11. 273.m = 4
  12. 274.m = 5
  13. 275.x = 15
  14. 276.x = 18
  15. 277.y = 12
  16. 278.y = 100
  17. 279.n = 0
  18. 280.n = 36
  19. 281.k = 2½ (also 5/2, 2.5)
  20. 282.k = 4½ (also 9/2, 4.5)
  21. 283.k = 2½ (also 5/2, 2.5)
  22. 284.k = 2⅓ (also 7/3, about 2.33)
  23. 285.k = ⅖ (also 0.4)
  24. 286.k = ½ (also 0.5)
  25. 287.t = 25
  26. 288.t = 12
  27. 289.w = 4½ (also 9/2, 4.5)
  28. 290.w = 7.5

15.Two steps (291–318)

  1. 291.x = 4
  2. 292.x = 7
  3. 293.x = 7
  4. 294.x = 5
  5. 295.y = 7
  6. 296.y = 7
  7. 297.y = 7
  8. 298.y = 0
  9. 299.n = 5
  10. 300.n = 5
  11. 301.m = 4
  12. 302.m = 9
  13. 303.k = 5
  14. 304.k = 10
  15. 305.x = 10
  16. 306.x = 14
  17. 307.x = 6
  18. 308.x = 10
  19. 309.a = 3
  20. 310.a = 4
  21. 311.t = 4
  22. 312.t = 5
  23. 313.b = 6
  24. 314.b = 10 (100 minus what is 80? It is 20. So 2b = 20.)
  25. 315.c = 2½ (also 5/2, 2.5)
  26. 316.c = 2½ (also 5/2, 2.5)
  27. 317.d = 5
  28. 318.d = 2

16.From words to an equation to the answer (319–334)

  1. 319.n + 6 = 15;  n = 9
  2. 320.n − 7 = 9;  n = 16
  3. 321.2n = 18;  n = 9
  4. 322.3n = 27;  n = 9
  5. 323.n/2 = 6;  n = 12
  6. 324.n/4 = 5;  n = 20
  7. 325.n + 4 = 11;  n = 7
  8. 326.n − 4 = 11;  n = 15
  9. 327.2n + 3 = 11;  n = 4
  10. 328.3n − 1 = 14;  n = 5
  11. 329.2n + 5 = 25;  n = 10
  12. 330.n/2 + 1 = 7;  n = 12
  13. 331.n + n = 30, which is 2n = 30;  n = 15
  14. 332.10 − n = 4;  n = 6
  15. 333.n + 9 = 9;  n = 0
  16. 334.2n + 8 = 20;  n = 6

17.Is it a solution? (335–346)

  1. 335.Yes. 4 + 5 = 9.
  2. 336.No. 5 − 3 = 2, not 8. The solution is x = 11.
  3. 337.Yes. 2 × 7 = 14.
  4. 338.No. 3 × 9 = 27, not 12. The solution is x = 4.
  5. 339.No. 3 ÷ 2 = 1.5, not 6. The solution is x = 12.
  6. 340.Yes. 2 × 4 + 1 = 9.
  7. 341.No. 3 × 3 − 2 = 7, not 10. The solution is x = 4.
  8. 342.Yes. 5 × 5 + 5 = 30.
  9. 343.Yes. 5 + 5 = 10.
  10. 344.Yes. 4 × 1 − 4 = 0.
  11. 345.No. 10 − 13 is below zero, not 3. The solution is x = 7.
  12. 346.Yes. 12 ÷ 4 + 1 = 3 + 1 = 4.

18.A little further (347–358)

  1. 347.x = 4
  2. 348.y = 5
  3. 349.n = 3
  4. 350.x = 5
  5. 351.y = 7
  6. 352.x = 5
  7. 353.x = 4
  8. 354.n = 5
  9. 355.m = 3
  10. 356.k = 5
  11. 357.x = 10
  12. 358.y = 5