Algebra worksheet for simplifying expressions, divided into five sections with progressively challenging problems.

Let's solve and simplify each expression step by step. The goal of simplifying expressions is to combine like terms — that means terms with the same variables raised to the same powers.

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🔷 Section A: Basic Addition

We combine like terms (variables and constants separately).

1) $ x + 3x + 4 + 3 = (1x + 3x) + (4 + 3) = 4x + 7 $

2) $ 5a + 2a + 8 + 6 = (5a + 2a) + (8 + 6) = 7a + 14 $

3) $ 7 + 2 + 6y + 3y = (7 + 2) + (6y + 3y) = 9 + 9y $

4) $ 8d + 6d + 7 + 1 = (8d + 6d) + (7 + 1) = 14d + 8 $

5) $ 3h + 2h + 7g + 9g = (3h + 2h) + (7g + 9g) = 5h + 16g $

6) $ 2x + 3 + 5x + 11 = (2x + 5x) + (3 + 11) = 7x + 14 $

7) $ 12 + 5r + 2 + 4r = (12 + 2) + (5r + 4r) = 14 + 9r $

8) $ 7y + 4 + y + 6 = (7y + y) + (4 + 6) = 8y + 10 $

9) $ 7u + 5v + u + v = (7u + u) + (5v + v) = 8u + 6v $

10) $ 9i + 4f + i + 3f = (9i + i) + (4f + 3f) = 10i + 7f $

11) $ 2c + 3d + 5c + e = (2c + 5c) + 3d + e = 7c + 3d + e $

12) $ 7 + 5x + 3 + 7y = (7 + 3) + 5x + 7y = 10 + 5x + 7y $

13) $ 9v + 2 + 14v + 8q = (9v + 14v) + 2 + 8q = 23v + 2 + 8q $

14) $ 3p + 4 + p + 3s + 5 = (3p + p) + (4 + 5) + 3s = 4p + 9 + 3s $

15) $ 7n + k + 4k + 8n + 3m = (7n + 8n) + (k + 4k) + 3m = 15n + 5k + 3m $

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🔷 Section B: Subtraction & Negative Terms

Combine like terms, remembering signs.

1) $ 2x - 6x = -4x $

2) $ 4a - 6a = -2a $

3) $ -4y + 3y = -y $

4) $ -2d + d = -d $

5) $ -3g + 5g = 2g $

6) $ -x + 2x = x $

7) $ 5r - 4r = r $

8) $ -7y - y = -8y $

9) $ -4v + 7v = 3v $

10) $ -2c + c = -c $

11) $ 5x - 9x = -4x $

12) $ -8v + 8v = 0 $

13) $ 3p - 6p = -3p $

14) $ -k + 3k = 2k $

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🔷 Section C: Mixed Operations

Combine like terms carefully.

1) $ 6x - x + 5 - 2 = (6x - x) + (5 - 2) = 5x + 3 $

2) $ 9a - 7a + 7 - 4 = (9a - 7a) + (7 - 4) = 2a + 3 $

3) $ 9 - 1 + 8y - 3y = (9 - 1) + (8y - 3y) = 8 + 5y $

4) $ 11d - 6d + 10 - 1 = (11d - 6d) + (10 - 1) = 5d + 9 $

5) $ 12h - 2h + 4g - 6g = (12h - 2h) + (4g - 6g) = 10h - 2g $

6) $ 2x - 5x + 1 - 4 = (-3x) + (-3) = -3x - 3 $

7) $ 2r - 4r + 6 - 10 = (-2r) + (-4) = -2r - 4 $

8) $ 7y + 4 - y + 8 = (7y - y) + (4 + 8) = 6y + 12 $

9) $ 7u + 5v - u + 6v = (7u - u) + (5v + 6v) = 6u + 11v $

10) $ 9i + 4f - i + 2f = (9i - i) + (4f + 2f) = 8i + 6f $

11) $ 4c + 3d - c + 4d = (4c - c) + (3d + 4d) = 3c + 7d $

12) $ 7 + 5x - 9 + 7x = (5x + 7x) + (7 - 9) = 12x - 2 $

13) $ 9v + 2 - 11v + 8 = (9v - 11v) + (2 + 8) = -2v + 10 $

14) $ 3p + 4s - 6p + 2s = (3p - 6p) + (4s + 2s) = -3p + 6s $

15) $ 7n + k + 4k - 8n = (7n - 8n) + (k + 4k) = -n + 5k $

16) $ 3x + 4y + 6x - 8y = (3x + 6x) + (4y - 8y) = 9x - 4y $

17) $ 7p - 4s + 2p + 2s = (7p + 2p) + (-4s + 2s) = 9p - 2s $

18) $ 5z + 8s - 9z + 9s = (5z - 9z) + (8s + 9s) = -4z + 17s $

19) $ -2a + 11b + 5a + b = (-2a + 5a) + (11b + b) = 3a + 12b $

20) $ -s + 4p + 6s + 2p = (-s + 6s) + (4p + 2p) = 5s + 6p $

21) $ -8j + 12s + 6j + 6s = (-8j + 6j) + (12s + 6s) = -2j + 18s $

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🔷 Section D: More Complex Expressions

Careful grouping.

1) $ x - 2y - 3x + 4y = (x - 3x) + (-2y + 4y) = -2x + 2y $

2) $ 5a - 2b - 8b + 6a = (5a + 6a) + (-2b - 8b) = 11a - 10b $

3) $ 7v + 2w - 6v - 3w = (7v - 6v) + (2w - 3w) = v - w $

4) $ 8d - 6e + 9e - 5d = (8d - 5d) + (-6e + 9e) = 3d + 3e $

5) $ 3g + 2h - 7g + 9g = (3g - 7g + 9g) + 2h = 5g + 2h $

6) $ 2x - 3 + 4x + 11 = (2x + 4x) + (-3 + 11) = 6x + 8 $

7) $ 12 - 5r - 2 + 4r = (12 - 2) + (-5r + 4r) = 10 - r $

8) $ 10y - 5z - y - 3z = (10y - y) + (-5z - 3z) = 9y - 8z $

9) $ 9u - v - 11u + v = (9u - 11u) + (-v + v) = -2u + 0 = -2u $

10) $ 15i - 4f - 17i + 3f = (15i - 17i) + (-4f + 3f) = -2i - f $

11) $ 16t - t - 5s - 9t = (16t - t - 9t) - 5s = 6t - 5s $

12) $ 19 - 5x - 8 + 7x = (19 - 8) + (-5x + 7x) = 11 + 2x $

13) $ 9v + 2v - 14v + 8q = (9v + 2v - 14v) + 8q = -3v + 8q $

14) $ 2c - 6d - 13c + 11e = (2c - 13c) - 6d + 11e = -11c - 6d + 11e $

15) $ 7n + k - 8k - 8n + 3m = (7n - 8n) + (k - 8k) + 3m = -n -7k + 3m $

16) $ 2x - 9y - 6x - 8y + 7z = (2x - 6x) + (-9y - 8y) + 7z = -4x - 17y + 7z $

17) $ 11p - 4s + 5r - 3p + 2s - r = (11p - 3p) + (-4s + 2s) + (5r - r) = 8p - 2s + 4r $

18) $ 5z - 8s - 9t + 9s + t = 5z + (-8s + 9s) + (-9t + t) = 5z + s - 8t $

19) $ a - c - 11b + a + b - 3c = (a + a) + (-c - 3c) + (-11b + b) = 2a - 4c - 10b $

20) $ -s + 4p + 6s + 2p - 5s = (-s + 6s - 5s) + (4p + 2p) = 0s + 6p = 6p $

21) $ 8j - 3 - 2s + 6j + 6s - 13 = (8j + 6j) + (-2s + 6s) + (-3 - 13) = 14j + 4s - 16 $

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🔷 Section E: Algebraic Terms with Powers

Only combine like terms — same variables and exponents.

1) $ xy + 3xy = 4xy $

2) $ 2ab + 4ab = 6ab $

3) $ 5st - st = 4st $

4) $ 4cd + 7c - 2cd = (4cd - 2cd) + 7c = 2cd + 7c $

5) $ y^2 + y^2 + y^2 = 3y^2 $

6) $ b^2 - b^2 - b^2 = (1 - 1 - 1)b^2 = -b^2 $

7) $ 4u^3 + 9u^3 = 13u^3 $

8) $ xy^2 + xy^2 = 2xy^2 $

9) $ 8xy^2 + xy - 2xy^2 - xy = (8xy^2 - 2xy^2) + (xy - xy) = 6xy^2 + 0 = 6xy^2 $

10) $ gh + h^2 + 3gh + 2h^2 = (gh + 3gh) + (h^2 + 2h^2) = 4gh + 3h^2 $

11) $ uv - v^2 + u^2 - uv - u = (uv - uv) + (-v^2) + u^2 - u = 0 + u^2 - v^2 - u = u^2 - u - v^2 $

12) $ 7ab^2 - 7a^2b + a^2b - 4ab^2 = (7ab^2 - 4ab^2) + (-7a^2b + a^2b) = 3ab^2 - 6a^2b $

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✔ Final Answers Summary

#### Section A:
1. $ 4x + 7 $
2. $ 7a + 14 $
3. $ 9 + 9y $
4. $ 14d + 8 $
5. $ 5h + 16g $
6. $ 7x + 14 $
7. $ 14 + 9r $
8. $ 8y + 10 $
9. $ 8u + 6v $
10. $ 10i + 7f $
11. $ 7c + 3d + e $
12. $ 10 + 5x + 7y $
13. $ 23v + 2 + 8q $
14. $ 4p + 9 + 3s $
15. $ 15n + 5k + 3m $

#### Section B:
1. $ -4x $
2. $ -2a $
3. $ -y $
4. $ -d $
5. $ 2g $
6. $ x $
7. $ r $
8. $ -8y $
9. $ 3v $
10. $ -c $
11. $ -4x $
12. $ 0 $
13. $ -3p $
14. $ 2k $

#### Section C:
1. $ 5x + 3 $
2. $ 2a + 3 $
3. $ 8 + 5y $
4. $ 5d + 9 $
5. $ 10h - 2g $
6. $ -3x - 3 $
7. $ -2r - 4 $
8. $ 6y + 12 $
9. $ 6u + 11v $
10. $ 8i + 6f $
11. $ 3c + 7d $
12. $ 12x - 2 $
13. $ -2v + 10 $
14. $ -3p + 6s $
15. $ -n + 5k $
16. $ 9x - 4y $
17. $ 9p - 2s $
18. $ -4z + 17s $
19. $ 3a + 12b $
20. $ 5s + 6p $
21. $ -2j + 18s $

#### Section D:
1. $ -2x + 2y $
2. $ 11a - 10b $
3. $ v - w $
4. $ 3d + 3e $
5. $ 5g + 2h $
6. $ 6x + 8 $
7. $ 10 - r $
8. $ 9y - 8z $
9. $ -2u $
10. $ -2i - f $
11. $ 6t - 5s $
12. $ 11 + 2x $
13. $ -3v + 8q $
14. $ -11c - 6d + 11e $
15. $ -n -7k + 3m $
16. $ -4x - 17y + 7z $
17. $ 8p - 2s + 4r $
18. $ 5z + s - 8t $
19. $ 2a - 4c - 10b $
20. $ 6p $
21. $ 14j + 4s - 16 $

#### Section E:
1. $ 4xy $
2. $ 6ab $
3. $ 4st $
4. $ 2cd + 7c $
5. $ 3y^2 $
6. $ -b^2 $
7. $ 13u^3 $
8. $ 2xy^2 $
9. $ 6xy^2 $
10. $ 4gh + 3h^2 $
11. $ u^2 - u - v^2 $
12. $ 3ab^2 - 6a^2b $

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📝 Explanation:

To simplify algebraic expressions, follow these steps:
1. Identify like terms — terms with the same variables and exponents.
2. Group like terms together.
3. Add or subtract coefficients of like terms.
4. Write the simplified expression in standard form (usually alphabetical order, with constants last).

For example:
- $ 3x + 2x = 5x $ → same variable
- $ 4y + 3y = 7y $
- But $ 3x + 4y $ cannot be combined

Also, remember:
- $ -x $ means $ -1x $
- $ 0 \times x = 0 $, so if all terms cancel, result is 0
- Constants (numbers without variables) are also like terms

This worksheet builds from simple addition to more complex combinations involving subtraction, multiple variables, and powers.

Let me know if you'd like this formatted as a printable answer sheet!