Algebra worksheet focusing on combining like terms and applying the distributive property to simplify expressions.

Problem Overview:

The worksheet focuses on two key algebraic skills:
1. Combining Like Terms: Simplifying expressions by adding or subtracting terms that have the same variable.
2. Using the Distributive Property: Expanding expressions by distributing a factor across terms inside parentheses.

We will solve each section step by step.

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Section 1: Combine Like Terms

#### Instructions:
Simplify each expression by combining like terms.

#### Solutions:
1. $ -4x + 5x $
- Combine the $ x $-terms: $ -4x + 5x = x $
- Answer: $ x $

2. $ 1 + 5v + v - 6 $
- Combine the constant terms: $ 1 - 6 = -5 $
- Combine the $ v $-terms: $ 5v + v = 6v $
- Answer: $ 6v - 5 $

3. $ 4n + 4 + 1 + 3n $
- Combine the $ n $-terms: $ 4n + 3n = 7n $
- Combine the constant terms: $ 4 + 1 = 5 $
- Answer: $ 7n + 5 $

4. $ 11a + 11a $
- Combine the $ a $-terms: $ 11a + 11a = 22a $
- Answer: $ 22a $

5. $ -2x - 8 - 7x + 2 $
- Combine the $ x $-terms: $ -2x - 7x = -9x $
- Combine the constant terms: $ -8 + 2 = -6 $
- Answer: $ -9x - 6 $

6. $ 7v + 6v $
- Combine the $ v $-terms: $ 7v + 6v = 13v $
- Answer: $ 13v $

7. $ -8x - 10x $
- Combine the $ x $-terms: $ -8x - 10x = -18x $
- Answer: $ -18x $

8. $ 6 - 7n - 2n - 8 $
- Combine the $ n $-terms: $ -7n - 2n = -9n $
- Combine the constant terms: $ 6 - 8 = -2 $
- Answer: $ -9n - 2 $

9. $ 2k - k $
- Combine the $ k $-terms: $ 2k - k = k $
- Answer: $ k $

10. $ -p - 11 + 3 $
- Combine the constant terms: $ -11 + 3 = -8 $
- Answer: $ -p - 8 $

11. $ 9n + 3n $
- Combine the $ n $-terms: $ 9n + 3n = 12n $
- Answer: $ 12n $

12. $ 12x + 11 - 4 $
- Combine the constant terms: $ 11 - 4 = 7 $
- Answer: $ 12x + 7 $

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Section 2: Use Distributive Property

#### Instructions:
Simplify each expression using the distributive property.

#### Solutions:
13. $ 3(-7 - 8n) $
- Distribute $ 3 $: $ 3 \cdot (-7) + 3 \cdot (-8n) = -21 - 24n $
- Answer: $ -21 - 24n $

14. $ -8(1 + 5m) $
- Distribute $ -8 $: $ -8 \cdot 1 + (-8) \cdot (5m) = -8 - 40m $
- Answer: $ -8 - 40m $

15. $ 8(r + 1) $
- Distribute $ 8 $: $ 8 \cdot r + 8 \cdot 1 = 8r + 8 $
- Answer: $ 8r + 8 $

16. $ 8(7x + 8) $
- Distribute $ 8 $: $ 8 \cdot 7x + 8 \cdot 8 = 56x + 64 $
- Answer: $ 56x + 64 $

17. $ 2(6n - 8) $
- Distribute $ 2 $: $ 2 \cdot 6n + 2 \cdot (-8) = 12n - 16 $
- Answer: $ 12n - 16 $

18. $ -3(8 - b) $
- Distribute $ -3 $: $ -3 \cdot 8 + (-3) \cdot (-b) = -24 + 3b $
- Answer: $ -24 + 3b $

19. $ -5(8v - 2) $
- Distribute $ -5 $: $ -5 \cdot 8v + (-5) \cdot (-2) = -40v + 10 $
- Answer: $ -40v + 10 $

20. $ -2(x - 5) $
- Distribute $ -2 $: $ -2 \cdot x + (-2) \cdot (-5) = -2x + 10 $
- Answer: $ -2x + 10 $

21. $ -(3a - 3) $
- Distribute $ -1 $: $ -1 \cdot 3a + (-1) \cdot (-3) = -3a + 3 $
- Answer: $ -3a + 3 $

22. $ -2(7 - 2n) $
- Distribute $ -2 $: $ -2 \cdot 7 + (-2) \cdot (-2n) = -14 + 4n $
- Answer: $ -14 + 4n $

23. $ -8(5 - 3v) $
- Distribute $ -8 $: $ -8 \cdot 5 + (-8) \cdot (-3v) = -40 + 24v $
- Answer: $ -40 + 24v $

24. $ -7(6x - 3) $
- Distribute $ -7 $: $ -7 \cdot 6x + (-7) \cdot (-3) = -42x + 21 $
- Answer: $ -42x + 21 $

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Section 3: Use Distributive Property, then Combine Like Terms

#### Instructions:
First, use the distributive property, then combine like terms to simplify each expression.

#### Solutions:
25. $ -n + 4(n + 1) $
- Distribute $ 4 $: $ -n + 4 \cdot n + 4 \cdot 1 = -n + 4n + 4 $
- Combine like terms: $ -n + 4n = 3n $
- Answer: $ 3n + 4 $

26. $ -3(1 - 3x) + 2x $
- Distribute $ -3 $: $ -3 \cdot 1 + (-3) \cdot (-3x) + 2x = -3 + 9x + 2x $
- Combine like terms: $ 9x + 2x = 11x $
- Answer: $ 11x - 3 $

27. $ -2(-3k + 4) - 7 $
- Distribute $ -2 $: $ -2 \cdot (-3k) + (-2) \cdot 4 - 7 = 6k - 8 - 7 $
- Combine constant terms: $ -8 - 7 = -15 $
- Answer: $ 6k - 15 $

28. $ -3p - (-8 + 4p) $
- Distribute $ -1 $: $ -3p - (-8) - 4p = -3p + 8 - 4p $
- Combine like terms: $ -3p - 4p = -7p $
- Answer: $ -7p + 8 $

29. $ -4 + 6(-4x + 3) $
- Distribute $ 6 $: $ -4 + 6 \cdot (-4x) + 6 \cdot 3 = -4 - 24x + 18 $
- Combine constant terms: $ -4 + 18 = 14 $
- Answer: $ -24x + 14 $

30. $ 3n + 3(1 + 8n) $
- Distribute $ 3 $: $ 3n + 3 \cdot 1 + 3 \cdot 8n = 3n + 3 + 24n $
- Combine like terms: $ 3n + 24n = 27n $
- Answer: $ 27n + 3 $

31. $ -2 + 5(4 + 3r) $
- Distribute $ 5 $: $ -2 + 5 \cdot 4 + 5 \cdot 3r = -2 + 20 + 15r $
- Combine constant terms: $ -2 + 20 = 18 $
- Answer: $ 15r + 18 $

32. $ -1 + 3(m + 4) $
- Distribute $ 3 $: $ -1 + 3 \cdot m + 3 \cdot 4 = -1 + 3m + 12 $
- Combine constant terms: $ -1 + 12 = 11 $
- Answer: $ 3m + 11 $

33. $ -(n + 2) - 2n $
- Distribute $ -1 $: $ -1 \cdot n + (-1) \cdot 2 - 2n = -n - 2 - 2n $
- Combine like terms: $ -n - 2n = -3n $
- Answer: $ -3n - 2 $

34. $ -3(5 + 2x) - 7 $
- Distribute $ -3 $: $ -3 \cdot 5 + (-3) \cdot 2x - 7 = -15 - 6x - 7 $
- Combine constant terms: $ -15 - 7 = -22 $
- Answer: $ -6x - 22 $

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Final Answers:

\[
\boxed{
\begin{array}{ll}
1. x & 2. 6v - 5 \\
3. 7n + 5 & 4. 22a \\
5. -9x - 6 & 6. 13v \\
7. -18x & 8. -9n - 2 \\
9. k & 10. -p - 8 \\
11. 12n & 12. 12x + 7 \\
13. -21 - 24n & 14. -8 - 40m \\
15. 8r + 8 & 16. 56x + 64 \\
17. 12n - 16 & 18. -24 + 3b \\
19. -40v + 10 & 20. -2x + 10 \\
21. -3a + 3 & 22. -14 + 4n \\
23. -40 + 24v & 24. -42x + 21 \\
25. 3n + 4 & 26. 11x - 3 \\
27. 6k - 15 & 28. -7p + 8 \\
29. -24x + 14 & 30. 27n + 3 \\
31. 15r + 18 & 32. 3m + 11 \\
33. -3n - 2 & 34. -6x - 22 \\
\end{array}
}
\]