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Problem: Combining Like Terms

The task is to simplify the given algebraic expressions by combining like terms. Like terms are terms that have the same variable raised to the same power. For example, \(3x\) and \(-7x\) are like terms, but \(3x\) and \(4y\) are not.

#### Step-by-Step Solutions:

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Expression 1: \(-6x - x\)

- Combine the \(x\)-terms: \(-6x - x = -7x\).
- Answer: \(\boxed{-7x}\)

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Expression 2: \(4x + 6x\)

- Combine the \(x\)-terms: \(4x + 6x = 10x\).
- Answer: \(\boxed{10x}\)

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Expression 3: \(3x - 7x\)

- Combine the \(x\)-terms: \(3x - 7x = -4x\).
- Answer: \(\boxed{-4x}\)

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Expression 4: \(-3x + 14x\)

- Combine the \(x\)-terms: \(-3x + 14x = 11x\).
- Answer: \(\boxed{11x}\)

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Expression 5: \(3 - 3x + 5x - 13\)

- Combine the \(x\)-terms: \(-3x + 5x = 2x\).
- Combine the constant terms: \(3 - 13 = -10\).
- Simplified expression: \(2x - 10\).
- Answer: \(\boxed{2x - 10}\)

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Expression 6: \(-6x + 7 + 2x - 2\)

- Combine the \(x\)-terms: \(-6x + 2x = -4x\).
- Combine the constant terms: \(7 - 2 = 5\).
- Simplified expression: \(-4x + 5\).
- Answer: \(\boxed{-4x + 5}\)

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Expression 7: \(3x + 6 + 4x + 7\)

- Combine the \(x\)-terms: \(3x + 4x = 7x\).
- Combine the constant terms: \(6 + 7 = 13\).
- Simplified expression: \(7x + 13\).
- Answer: \(\boxed{7x + 13}\)

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Expression 8: \(2x + 11 - 7x - 3\)

- Combine the \(x\)-terms: \(2x - 7x = -5x\).
- Combine the constant terms: \(11 - 3 = 8\).
- Simplified expression: \(-5x + 8\).
- Answer: \(\boxed{-5x + 8}\)

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Expression 9: \(6x + 3 - 5x + 4\)

- Combine the \(x\)-terms: \(6x - 5x = x\).
- Combine the constant terms: \(3 + 4 = 7\).
- Simplified expression: \(x + 7\).
- Answer: \(\boxed{x + 7}\)

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Expression 10: \(7x + 8 + x - 9\)

- Combine the \(x\)-terms: \(7x + x = 8x\).
- Combine the constant terms: \(8 - 9 = -1\).
- Simplified expression: \(8x - 1\).
- Answer: \(\boxed{8x - 1}\)

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Expression 11: \(-10x - x + 4 + 6\)

- Combine the \(x\)-terms: \(-10x - x = -11x\).
- Combine the constant terms: \(4 + 6 = 10\).
- Simplified expression: \(-11x + 10\).
- Answer: \(\boxed{-11x + 10}\)

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Expression 12: \(8x + 3 - 6 + 9\)

- Combine the \(x\)-terms: There is only one \(x\)-term, \(8x\).
- Combine the constant terms: \(3 - 6 + 9 = 6\).
- Simplified expression: \(8x + 6\).
- Answer: \(\boxed{8x + 6}\)

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Expression 13: \(12x - 22 + 12 - 7x\)

- Combine the \(x\)-terms: \(12x - 7x = 5x\).
- Combine the constant terms: \(-22 + 12 = -10\).
- Simplified expression: \(5x - 10\).
- Answer: \(\boxed{5x - 10}\)

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Expression 14: \(7x + 8y - 3x\)

- Combine the \(x\)-terms: \(7x - 3x = 4x\).
- The \(y\)-term remains as it is: \(8y\).
- Simplified expression: \(4x + 8y\).
- Answer: \(\boxed{4x + 8y}\)

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Expression 15: \(3x + 4y + 8x - 3y\)

- Combine the \(x\)-terms: \(3x + 8x = 11x\).
- Combine the \(y\)-terms: \(4y - 3y = y\).
- Simplified expression: \(11x + y\).
- Answer: \(\boxed{11x + y}\)

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Expression 16: \(13x + 2x - 4x + 8 + 3\)

- Combine the \(x\)-terms: \(13x + 2x - 4x = 11x\).
- Combine the constant terms: \(8 + 3 = 11\).
- Simplified expression: \(11x + 11\).
- Answer: \(\boxed{11x + 11}\)

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Final Answers:

1. \(\boxed{-7x}\)
2. \(\boxed{10x}\)
3. \(\boxed{-4x}\)
4. \(\boxed{11x}\)
5. \(\boxed{2x - 10}\)
6. \(\boxed{-4x + 5}\)
7. \(\boxed{7x + 13}\)
8. \(\boxed{-5x + 8}\)
9. \(\boxed{x + 7}\)
10. \(\boxed{8x - 1}\)
11. \(\boxed{-11x + 10}\)
12. \(\boxed{8x + 6}\)
13. \(\boxed{5x - 10}\)
14. \(\boxed{4x + 8y}\)
15. \(\boxed{11x + y}\)
16. \(\boxed{11x + 11}\)

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Boxed Final Answer:
\[
\boxed{
\begin{array}{ll}
1. & -7x \\
2. & 10x \\
3. & -4x \\
4. & 11x \\
5. & 2x - 10 \\
6. & -4x + 5 \\
7. & 7x + 13 \\
8. & -5x + 8 \\
9. & x + 7 \\
10. & 8x - 1 \\
11. & -11x + 10 \\
12. & 8x + 6 \\
13. & 5x - 10 \\
14. & 4x + 8y \\
15. & 11x + y \\
16. & 11x + 11 \\
\end{array}
}
\]