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? \( P=3 S-15 \) \( P \cdot 5=3 S-5 \)

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 \( \boxed{10} \).