Let's solve the equation:

\[
\frac{x}{4} + \frac{6}{7}(6x + 7) = 610
\]

### Step 1: Distribute the \(\frac{6}{7}\) across \((6x + 7)\)
\[
\frac{x}{4} + \frac{6}{7}(6x) + \frac{6}{7}(7) = 610
\]

\[
\frac{x}{4} + \frac{36x}{7} + 6 = 610
\]

### Step 2: Move constants to the other side
\[
\frac{x}{4} + \frac{36x}{7} = 610 - 6
\]
\[
\frac{x}{4} + \frac{36x}{7} = 604
\]

### Step 3: Find a common denominator
The common denominator for 4 and 7 is 28. So, rewrite both fractions:
\[
\frac{x}{4} = \frac{7x}{28}, \quad \frac{36x}{7} = \frac{144x}{28}
\]
Now the equation becomes:
\[
\frac{7x}{28} + \frac{144x}{28} = 604
\]

### Step 4: Combine the terms
\[
\frac{7x + 144x}{28} = 604
\]
\[
\frac{151x}{28} = 604
\]

### Step 5: Multiply both sides by 28 to eliminate the denominator
\[
151x = 604 \times 28
\]
\[
151x = 16912
\]

### Step 6: Solve for \(x\)
\[
x = \frac{16912}{151}
\]
\[
x = 112
\]

### Final Answer:
\[
x = 112
\]

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. Solve \(\frac{3x}{5} + \frac{7}{10}(5x + 3) = 50\).
2. Simplify \(\frac{5}{6}(2x + 3) + \frac{4x}{3} = 40\).
3. Solve \(\frac{2x}{3} + \frac{5}{6}(6x + 9) = 90\).
4. Solve \(\frac{x}{5} + \frac{7x}{6} = 65\).
5. Solve \(\frac{4x}{9} + \frac{3}{8}(8x + 9) = 100\).

**Tip**: Always check if you can simplify fractions or factor out common terms to make calculations easier.

Learn how to solve the linear equation x/4 + 6/7(6x + 7) = 610 with a detailed step-by-step solution. This problem covers algebraic concepts like fraction operations, the distributive property, and finding common denominators. Suitable for students in Grades 9-10.

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