https://file.aiquickdraw.com/mathbot/file/1727523633991k5zwzrij.jpg

The first equation given in the image is:

\[
\frac{8x - 4}{4} + 7 = \frac{4x + 4}{6}
\]

We will solve this step by step for \(x\).

### Step 1: Simplify both sides of the equation.
The left-hand side (LHS):

\[
\frac{8x - 4}{4} + 7 = 2x - 1 + 7 = 2x + 6
\]

The right-hand side (RHS):

\[
\frac{4x + 4}{6} = \frac{4(x + 1)}{6} = \frac{2}{3}(x + 1)
\]

So, the equation becomes:

\[
2x + 6 = \frac{2}{3}(x + 1)
\]

### Step 2: Eliminate the fraction by multiplying both sides by 3.
\[
3(2x + 6) = 2(x + 1)
\]

Simplifying:

\[
6x + 18 = 2x + 2
\]

### Step 3: Solve for \(x\).
Subtract \(2x\) from both sides:

\[
4x + 18 = 2
\]

Subtract 18 from both sides:

\[
4x = -16
\]

Divide by 4:

\[
x = -4
\]

Thus, \(x = -4\), which corresponds to option **b**.

---

The second equation is:

\[
69(57a + 92) - 21 = 41724
\]

Let's solve this for \(a\).

### Step 1: Distribute 69.
\[
69(57a + 92) = 69 \cdot 57a + 69 \cdot 92 = 3933a + 6348
\]

So the equation becomes:

\[
3933a + 6348 - 21 = 41724
\]

Simplify:

\[
3933a + 6327 = 41724
\]

### Step 2: Subtract 6327 from both sides.
\[
3933a = 35397
\]

### Step 3: Solve for \(a\).
\[
a = \frac{35397}{3933} = 9
\]

Thus, \(a = 9\).

---

Let me know if you'd like further details or if you have any questions.

Here are some related questions:
1. How can we solve similar equations with fractions?
2. What are the common strategies for eliminating fractions in equations?
3. How do we verify the solutions of linear equations?
4. How does distributing multiplication help in solving equations?
5. What are other methods to solve linear equations without distributing first?

**Tip:** Always check for simpler forms of expressions before proceeding with the solution to save time.

Solve the equation (8x - 4) / 4 + 7 = (4x + 4) / 6 for x. Also, solve 69(57a + 92) - 21 = 41724 for a.

Step-by-step solutions for solving the linear equation (8x - 4) / 4 + 7 = (4x + 4) / 6 for x and the equation 69(57a + 92) - 21 = 41724 for a. This guide covers key algebraic concepts such as simplifying expressions, eliminating fractions, and solving linear equations. Suitable for high school students in Grades 9-12.

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