To solve the equation \( \frac{7x - 8}{9x + 4} = \frac{7x + 1}{9x + 8} \), follow these steps:

### Step 1: Cross-multiply
We'll use cross-multiplication since we have a rational equation:

\[
(7x - 8)(9x + 8) = (7x + 1)(9x + 4)
\]

### Step 2: Expand both sides

#### Left-hand side:
\[
(7x - 8)(9x + 8) = 7x(9x) + 7x(8) - 8(9x) - 8(8)
= 63x^2 + 56x - 72x - 64
= 63x^2 - 16x - 64
\]

#### Right-hand side:
\[
(7x + 1)(9x + 4) = 7x(9x) + 7x(4) + 1(9x) + 1(4)
= 63x^2 + 28x + 9x + 4
= 63x^2 + 37x + 4
\]

### Step 3: Set the equation equal to zero

Now that both sides are expanded, we can set the equation equal to zero by subtracting the right-hand side from the left-hand side:

\[
(63x^2 - 16x - 64) - (63x^2 + 37x + 4) = 0
\]

Simplify:

\[
63x^2 - 16x - 64 - 63x^2 - 37x - 4 = 0
\]
\[
-16x - 37x - 64 - 4 = 0
\]
\[
-53x - 68 = 0
\]

### Step 4: Solve for \( x \)

Now solve for \( x \):

\[
-53x = 68
\]
\[
x = \frac{68}{-53}
\]
\[
x = -\frac{68}{53}
\]

Thus, the solution is:

\[
x = -\frac{68}{53}
\]

Would you like more details or have any questions?

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Here are five related questions:
1. How do you handle a rational equation with different denominators?
2. What are the steps involved in solving a quadratic equation?
3. Can you explain the process of cross-multiplication in rational equations?
4. How do you verify the solution of a rational equation?
5. What are common mistakes to avoid when solving rational equations?

**Tip:** Always check for extraneous solutions when solving rational equations by substituting the solution back into the original equation.

Learn how to solve the rational equation (7x - 8)/(9x + 4) = (7x + 1)/(9x + 8) using cross-multiplication. This detailed solution breaks down each step, including expanding both sides, simplifying the equation, and solving for x. Suitable for students in Grades 9-12.

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