Age Problem Explained: System of Equations Method

This problem involves finding the sum of the ages of two people, A and B, based on two given conditions relating their ages. We can solve this using a system of linear equations.

Setting Up the Age Equations

Let's represent the age of A as '$A$' and the age of B as '$B$'. We can translate the given information into two equations:

• "The sum of the age of A and 3 times the age of B is 43 years." This translates to:

  $A + 3B = 43 \quad (1)$

• "When 5 times the age of A is added to 2 times the age of B, the result is 98 years." This translates to:

  $5A + 2B = 98 \quad (2)$

Solving the System of Equations

We need to solve these two equations simultaneously to find the values of '$A$' and '$B$'. We can use the elimination method.

1. Multiply Equation (1) to match A coefficients: Multiply the first equation by 5 so that the coefficient of '$A$' matches in both equations.

   $5 \times (A + 3B) = 5 \times 43$

   $5A + 15B = 215 \quad (3)$

2. Eliminate A: Subtract Equation (2) from the new Equation (3).

   $(5A + 15B) - (5A + 2B) = 215 - 98$

   $5A + 15B - 5A - 2B = 117$

   $13B = 117$

3. Solve for B: Divide both sides by 13.

   $B = \frac{117}{13}$

   $B = 9$

   So, the age of B is 9 years.

4. Substitute B to find A: Substitute the value of '$B$' (which is 9) back into Equation (1).

   $A + 3(9) = 43$

   $A + 27 = 43$

5. Solve for A: Subtract 27 from both sides.

   $A = 43 - 27$

   $A = 16$

   So, the age of A is 16 years.

Calculating the Sum of Ages

The question asks for the sum of the ages of A and B.

Sum = Age of A + Age of B

Sum = $A + B$

Sum = $16 + 9$

Sum = $25$

The sum of the ages of A and B is 25 years.