This problem involves setting up and solving a system of linear equations based on the information given about the ages of two people, A and B.

Age Word Problem Setup

First, let's define variables for the unknown ages:

• Let $A$ be the current age of person A.
• Let $B$ be the current age of person B.

Now, we translate the sentences from the question into mathematical equations:

1. "The sum of the age of A and 5 times the age of B is 55 years." This translates to:

   $ A + 5B = 55 \quad (Equation \, 1) $

2. "When 3 times the age of A is added to 7 times the age of B, the result is 99 years." This translates to:

   $ 3A + 7B = 99 \quad (Equation \, 2) $

Our goal is to find the sum of their ages, which is $A + B$. To do this, we first need to find the individual values of $A$ and $B$ by solving the system of equations.

Solving the System of Linear Equations

We can use the elimination method to solve the system:

$ A + 5B = 55 \quad (1) $

$ 3A + 7B = 99 \quad (2) $

To eliminate $A$, we can multiply Equation 1 by 3:

$ 3 \times (A + 5B) = 3 \times 55 $

$ 3A + 15B = 165 \quad (Equation \, 3) $

Now, subtract Equation 2 from Equation 3:

$ (3A + 15B) - (3A + 7B) = 165 - 99 $

$ 3A + 15B - 3A - 7B = 66 $

$ 8B = 66 $

Solve for $B$:

$ B = \frac{66}{8} $

Simplify the fraction:

$ B = \frac{33}{4} $

Now, substitute the value of $B$ back into Equation 1 to find $A$:

$ A + 5B = 55 $

$ A + 5 \left( \frac{33}{4} \right) = 55 $

$ A + \frac{165}{4} = 55 $

Solve for $A$:

$ A = 55 - \frac{165}{4} $

To subtract, find a common denominator:

$ A = \frac{55 \times 4}{4} - \frac{165}{4} $

$ A = \frac{220}{4} - \frac{165}{4} $

$ A = \frac{220 - 165}{4} $

$ A = \frac{55}{4} $

Calculating the Sum of Ages

We have found the ages:

• $A = \frac{55}{4}$ years
• $B = \frac{33}{4}$ years

The question asks for the sum of the ages, $A + B$.

$ A + B = \frac{55}{4} + \frac{33}{4} $

Since the denominators are the same, we can add the numerators:

$ A + B = \frac{55 + 33}{4} $

$ A + B = \frac{88}{4} $

$ A + B = 22 $

So, the sum of the ages of A and B is 22 years.