**Problem Analysis:**

Let's represent the tens digit as 'x' and the ones digit as 'y'. So the two-digit number can be represented as 10x + y.

According to the problem, the sum of the digits is 10. So we have the equation:

x + y = 10

Now, if we subtract 18 from the two-digit number, we get:

(10x + y) - 18

We need to find a two-digit number for which the digits of the resulting number are equal. Let's represent this digit as 'z'. So we have the equation:

(10x + y) - 18 = z

**Solution:**

To solve this problem, we will use the given equations and find the values of x and y.

1. **Using the first equation:**

x + y = 10

Rearranging the equation, we get:

x = 10 - y

2. **Substituting x in the second equation:**

(10x + y) - 18 = z

Substituting the value of x from the first equation, we get:

(10(10 - y) + y) - 18 = z

Simplifying the equation, we get:

100 - 10y + y - 18 = z

Combining like terms, we get:

82 - 9y = z

Since the digits in the resulting number are equal, we have:

y = z

Substituting this value in the equation, we get:

82 - 9y = y

Rearranging the equation, we get:

82 = 10y

Dividing both sides by 10, we get:

8.2 = y

Since y represents the ones digit, it cannot be a decimal. Therefore, there is no solution for this equation.

**Conclusion:**

Based on our analysis, there is no two-digit number for which the digits of the resulting number are equal.