The sum of Pete's and Sam's ages is 30 . Five years ago, Pete was 3 times as old as Sam. How old is Sam? Let $$ P= $$ Pete's age, $$ S $$ = Sam's age, and $$ P+S=30 $$. Which of the following equations would complete the system? $$ P=3 S-15 $$ $$ P \cdot 5=3 S-5 $$

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To solve the problem, we start by extracting the known conditions and setting up the equations based on the information provided.

1. **Known Conditions:**
   - The sum of Pete's and Sam's ages is 30: 
     $$
     P + S = 30
     $$
   - Five years ago, Pete was 3 times as old as Sam:
     - Five years ago, Pete's age was $$ P - 5 $$ and Sam's age was $$ S - 5 $$.
     - The relationship can be expressed as:
     $$
     P - 5 = 3(S - 5)
     $$

2. **Setting Up the Equations:**
   - We already have the first equation: 
     $$
     P + S = 30
     $$
   - Now, let's simplify the second equation:
     $$
     P - 5 = 3(S - 5)
     $$
   - Expanding the right side:
     $$
     P - 5 = 3S - 15
     $$
   - Rearranging gives:
     $$
     P = 3S - 10
     $$

3. **Identifying the Correct Equation:**
   - We need to compare the derived equation $$ P = 3S - 10 $$ with the options provided:
     - $$ P = 3S - 15 $$ (This is incorrect because it does not match our derived equation.)
     - $$ P \cdot 5 = 3S - 5 $$ (This is also incorrect as it does not represent the relationship correctly.)

4. **Conclusion:**
   - Neither of the provided equations correctly completes the system based on the information given. The correct equation derived from the conditions is $$ P = 3S - 10 $$.

Now, let's solve the system of equations to find Sam's age $$ S $$.

We have:
1. $$ P + S = 30 $$
2. $$ P = 3S - 10 $$

Substituting the second equation into the first:
$$
(3S - 10) + S = 30
$$
Combining like terms:
$$
4S - 10 = 30
$$
Adding 10 to both sides:
$$
4S = 40
$$
Dividing by 4:
$$
S = 10
$$

Thus, Sam's age is $$ S = 10 $$.

To summarize:
- Sam's age is $$ \boxed{10} $$.
Sam is 10 years old.

The sum of Pete's and Sam's ages is 30 . Five years ago, Pete was 3 times as old as Sam. How old is Sam? Let \( P= \) Pete's age, \( S \) = Sam's age, and \( P+S=30 \). Which of the following equations would complete the system? \( P=3 S-15 \) \( P \cdot 5=3 S-5 \)

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