Solving the Father's Age Problem

Let the son's age be represented by \(S\) and the father's age by \(F\). We can set up a system of two linear equations based on the information given:

• Equation 1: "2 times a son's age is added to his father's age, the sum is 46 years." $2S + F = 46$
• Equation 2: "2.5 times the father's age is added to the son's age, the sum is 97 years." $S + 2.5F = 97$

Algebraic Solution Steps

We need to solve this system of equations to find the value of \(F\) (father's age).

1. Isolate Son's Age (S) in Equation 1:
   From \(2S + F = 46\), we get \(2S = 46 - F\).
   Therefore, \(S = \frac{46 - F}{2}\).

2. Substitute S in Equation 2:
   Replace \(S\) in the second equation with the expression derived above:
   $ \left( \frac{46 - F}{2} \right) + 2.5F = 97 $

3. Solve for Father's Age (F):
   To eliminate the fraction, multiply the entire equation by 2:
   $ 2 \left( \frac{46 - F}{2} \right) + 2(2.5F) = 2(97) $
   $ (46 - F) + 5F = 194 $
   Combine the terms involving \(F\):
   $ 46 + 4F = 194 $
   Subtract 46 from both sides:
   $ 4F = 194 - 46 $
   $ 4F = 148 $
   Divide by 4:
   $ F = \frac{148}{4} $
   $ F = 37 $

The father's age is 37 years.