https://file.aiquickdraw.com/mathbot/file/1727719538304ypnfcsca.jpg

Let's go through the problems in the image step by step.

### 1. **Simplify the following expressions using the distributive property:**

#### Question 7:
\[ \frac{1}{2}(8 - 6j) \]

Apply the distributive property:
\[
\frac{1}{2} \times 8 = 4, \quad \frac{1}{2} \times -6j = -3j
\]
So, the simplified expression is:
\[
4 - 3j
\]

#### Question 8:
\[ 8(p + 7 + 2q) \]

Distribute the 8:
\[
8 \times p = 8p, \quad 8 \times 7 = 56, \quad 8 \times 2q = 16q
\]
The simplified expression is:
\[
8p + 56 + 16q
\]

#### Question 9:
\[ 4(3.2d - 10) \]

Distribute the 4:
\[
4 \times 3.2d = 12.8d, \quad 4 \times -10 = -40
\]
The simplified expression is:
\[
12.8d - 40
\]

---

### 2. **Simplify the expressions using the distributive property. Then, combine any like terms:**

#### Question 10:
\[ 6(4c - 2) + 10 \]

First, distribute the 6:
\[
6 \times 4c = 24c, \quad 6 \times -2 = -12
\]
Now, the expression becomes:
\[
24c - 12 + 10
\]
Combine like terms:
\[
24c - 2
\]

#### Question 11:
\[ 6(3b + 7) + 5b \]

First, distribute the 6:
\[
6 \times 3b = 18b, \quad 6 \times 7 = 42
\]
Now, the expression becomes:
\[
18b + 42 + 5b
\]
Combine like terms:
\[
(18b + 5b) + 42 = 23b + 42
\]

#### Question 12:
\[ 3x + 2(4x + 7) \]

First, distribute the 2:
\[
2 \times 4x = 8x, \quad 2 \times 7 = 14
\]
Now, the expression becomes:
\[
3x + 8x + 14
\]
Combine like terms:
\[
(3x + 8x) + 14 = 11x + 14
\]

---

### 3. **Factoring an Expression:**

- Expressions can be **factored** to remove the common factor. This is how we "reverse" the distributive property.
- Steps:
  1. Determine the greatest common factor (GCF) between the values.
  2. Remove the GCF by dividing each term by it.

The example provided:
\[
6x + 15; \quad \text{GCF is 3}, \quad \text{factored form: } 3(2x + 5)
\]

#### Area Model (GCF for \(4x + 16\)):

Find the GCF of \(4x\) and 16. The GCF is 4. So, factoring out the GCF:
\[
4x + 16 = 4(x + 4)
\]

---

### 4. **Rewriting by factoring out the GCF:**

#### Question 13:
\[ 6a + 2 \]

The GCF of 6 and 2 is 2. So, factor out the GCF:
\[
6a + 2 = 2(3a + 1)
\]

#### Question 14:
\[ 24x - 9 \]

The GCF of 24 and 9 is 3. So, factor out the GCF:
\[
24x - 9 = 3(8x - 3)
\]

#### Question 15:
\[ 10x + 35 \]

The GCF of 10 and 35 is 5. So, factor out the GCF:
\[
10x + 35 = 5(2x + 7)
\]

---

Let me know if you'd like more details on any specific problem! Here are some additional related questions for further practice:

1. How do you determine the greatest common factor (GCF) of two terms?
2. What is the purpose of factoring an expression in algebra?
3. How does the distributive property work when there are variables involved?
4. Can you always use the distributive property in reverse to factor an expression? Why or why not?
5. What is the next step after factoring out the GCF when solving an equation?

**Tip:** When factoring an expression, always double-check by distributing back to verify your answer!

Simplify expressions using the distributive property and factor expressions using GCF.

Learn how to simplify expressions using the distributive property and factor expressions by identifying the greatest common factor (GCF). This guide provides step-by-step solutions for combining like terms, factoring expressions, and simplifying algebraic equations. Suitable for students in Grades 7-9.

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