We are given the equation:

\[
\frac{4}{9}\left(\frac{3}{8}x - \frac{27}{16}\right) = \frac{13}{4}
\]

### Step 1: Distribute \(\frac{4}{9}\)
We distribute \(\frac{4}{9}\) across the terms inside the parentheses:

\[
\frac{4}{9} \cdot \frac{3}{8}x - \frac{4}{9} \cdot \frac{27}{16} = \frac{13}{4}
\]

Multiplying the fractions:

\[
\frac{4}{9} \cdot \frac{3}{8}x = \frac{12}{72}x = \frac{1}{6}x
\]
\[
\frac{4}{9} \cdot \frac{27}{16} = \frac{108}{144} = \frac{3}{4}
\]

So the equation becomes:

\[
\frac{1}{6}x - \frac{3}{4} = \frac{13}{4}
\]

### Step 2: Eliminate fractions
To eliminate the fractions, we can multiply the entire equation by 12, the least common denominator of 6 and 4:

\[
12\left(\frac{1}{6}x - \frac{3}{4}\right) = 12 \cdot \frac{13}{4}
\]

This simplifies to:

\[
2x - 9 = 39
\]

### Step 3: Solve for \(x\)
Now, solve for \(x\):

\[
2x = 39 + 9
\]
\[
2x = 48
\]
\[
x = \frac{48}{2} = 24
\]

### Final answer:
\[
x = 24
\]

Would you like more details or have any questions?

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Here are 5 related questions to extend this concept:
1. What is the process for solving equations with different denominators?
2. How do you verify the solution to an equation?
3. What is the best way to handle fractions in algebraic equations?
4. Can you solve a similar equation with different coefficients?
5. How would you approach a system of equations that involves fractions?

**Tip:** When dealing with fractions, try to eliminate them early by multiplying through by the least common denominator.

Learn how to solve the linear equation 4/9(3/8x - 27/16) = 13/4 by distributing, eliminating fractions, and solving for x. This problem covers key algebraic concepts, including the distributive property and handling equations with fractions. Suitable for students in Grades 8-10.

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