Use this worksheet to practice applying the distributive property to simplify algebraic expressions involving positive and negative integers.

Problem: Simplify the given expressions using the Distributive Property.

The distributive property states:
$$
a(b + c) = ab + ac
$$
This means that you multiply the term outside the parentheses by each term inside the parentheses and then add the results.

Let's solve each problem step by step.

---

Problem 1: \( -5 \cdot (10 + a) \)

1. Distribute \(-5\) to both terms inside the parentheses:
$$
-5 \cdot (10 + a) = (-5 \cdot 10) + (-5 \cdot a)
$$

2. Simplify each term:
$$
(-5 \cdot 10) = -50 \quad \text{and} \quad (-5 \cdot a) = -5a
$$

3. Combine the results:
$$
-50 - 5a
$$

Final Answer:
$$
\boxed{-50 - 5a}
$$

---

Problem 2: \( 5(x + 2) - 8x \)

1. Distribute \(5\) to both terms inside the parentheses:
$$
5(x + 2) = (5 \cdot x) + (5 \cdot 2) = 5x + 10
$$

2. Substitute back into the expression:
$$
5(x + 2) - 8x = (5x + 10) - 8x
$$

3. Combine like terms:
$$
5x - 8x + 10 = -3x + 10
$$

Final Answer:
$$
\boxed{-3x + 10}
$$

---

Problem 3: \( -(8 - 2a) \cdot 5 \)

1. Distribute the negative sign inside the parentheses:
$$
-(8 - 2a) = -8 + 2a
$$

2. Now distribute \(5\) to both terms:
$$
5(-8 + 2a) = (5 \cdot -8) + (5 \cdot 2a) = -40 + 10a
$$

Final Answer:
$$
\boxed{-40 + 10a}
$$

---

Problem 4: \( 5(1 - 5a) - 1 \)

1. Distribute \(5\) to both terms inside the parentheses:
$$
5(1 - 5a) = (5 \cdot 1) + (5 \cdot -5a) = 5 - 25a
$$

2. Substitute back into the expression:
$$
5(1 - 5a) - 1 = (5 - 25a) - 1
$$

3. Combine like terms:
$$
5 - 1 - 25a = 4 - 25a
$$

Final Answer:
$$
\boxed{4 - 25a}
$$

---

Problem 5: \( 2 - 6(a + 8) \)

1. Distribute \(-6\) to both terms inside the parentheses:
$$
-6(a + 8) = (-6 \cdot a) + (-6 \cdot 8) = -6a - 48
$$

2. Substitute back into the expression:
$$
2 - 6(a + 8) = 2 + (-6a - 48)
$$

3. Combine like terms:
$$
2 - 48 - 6a = -46 - 6a
$$

Final Answer:
$$
\boxed{-46 - 6a}
$$

---

Problem 6: \( 7x - (9 + 6x) \)

1. Distribute the negative sign inside the parentheses:
$$
-(9 + 6x) = -9 - 6x
$$

2. Substitute back into the expression:
$$
7x - (9 + 6x) = 7x - 9 - 6x
$$

3. Combine like terms:
$$
7x - 6x - 9 = x - 9
$$

Final Answer:
$$
\boxed{x - 9}
$$

---

Problem 7: \( -(61 - 8) + 5r \)

1. Simplify the expression inside the parentheses:
$$
61 - 8 = 53
$$

2. Distribute the negative sign:
$$
-(61 - 8) = -53
$$

3. Substitute back into the expression:
$$
-(61 - 8) + 5r = -53 + 5r
$$

Final Answer:
$$
\boxed{-53 + 5r}
$$

---

Problem 8: \( -7(1 - 6a) + 7a \)

1. Distribute \(-7\) to both terms inside the parentheses:
$$
-7(1 - 6a) = (-7 \cdot 1) + (-7 \cdot -6a) = -7 + 42a
$$

2. Substitute back into the expression:
$$
-7(1 - 6a) + 7a = (-7 + 42a) + 7a
$$

3. Combine like terms:
$$
-7 + 42a + 7a = -7 + 49a
$$

Final Answer:
$$
\boxed{-7 + 49a}
$$

---

Problem 9: \( 9a - 2(1 + 2a) \)

1. Distribute \(-2\) to both terms inside the parentheses:
$$
-2(1 + 2a) = (-2 \cdot 1) + (-2 \cdot 2a) = -2 - 4a
$$

2. Substitute back into the expression:
$$
9a - 2(1 + 2a) = 9a + (-2 - 4a)
$$

3. Combine like terms:
$$
9a - 4a - 2 = 5a - 2
$$

Final Answer:
$$
\boxed{5a - 2}
$$

---

Problem 10: \( 4 + 5(3 + 8a) \)

1. Distribute \(5\) to both terms inside the parentheses:
$$
5(3 + 8a) = (5 \cdot 3) + (5 \cdot 8a) = 15 + 40a
$$

2. Substitute back into the expression:
$$
4 + 5(3 + 8a) = 4 + (15 + 40a)
$$

3. Combine like terms:
$$
4 + 15 + 40a = 19 + 40a
$$

Final Answer:
$$
\boxed{19 + 40a}
$$

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Final Answers:

1. \( \boxed{-50 - 5a} \)
2. \( \boxed{-3x + 10} \)
3. \( \boxed{-40 + 10a} \)
4. \( \boxed{4 - 25a} \)
5. \( \boxed{-46 - 6a} \)
6. \( \boxed{x - 9} \)
7. \( \boxed{-53 + 5r} \)
8. \( \boxed{-7 + 49a} \)
9. \( \boxed{5a - 2} \)
10. \( \boxed{19 + 40a} \)