A man's age is given by the expression $3p + 2q$ years, where his only son's age is $p$ years and his only daughter's age is $q$ years. The son is 3 years older than his sister. The man's wife is 5 years younger than him, and the man is 34 years old. What is the average age of the family?

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**Solution:**

1. According to the question, the man's age is $34$ years.

2. The man's wife is $5$ years younger than him, so her age is:
   $$34 - 5 = 29 	ext{ years}$$

3. Given the man's age is expressed as:
   $$3p + 2q = 34 \quad 	ext{(Equation 1)}$$

4. The son is $3$ years older than the daughter, so:
   $$p - q = 3 \quad 	ext{(Equation 2)}$$

5. From Equation 2, express $p$ in terms of $q$:
   $$p = q + 3$$

6. Substitute $p = q + 3$ into Equation 1:
   $$3(q + 3) + 2q = 34$$
   $$3q + 9 + 2q = 34$$
   $$5q + 9 = 34$$
   $$5q = 25$$
   $$q = 5$$

7. Find $p$:
   $$p = 5 + 3 = 8$$

8. Now, calculate the average age of the family (man, wife, son, daughter):
   $$\frac{34 + 29 + 8 + 5}{4} = \frac{76}{4} = 19$$

**Answer:** $19.0$

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