**Problem Analysis:**

Let's break down the information given in the problem:

1. The sum of the digits of a two-digit number is 10.
2. If 18 is subtracted from this number, the resulting number will have equal digits.

We need to find the two-digit number that satisfies these conditions.

**Solution:**

Let's assume the two-digit number to be xy.

According to the given information, we can form the following equations:

1. x + y = 10 (Equation 1) (since the sum of the digits is 10)
2. 10x + y - 18 = 10y + x (Equation 2) (since the resulting number after subtracting 18 will have equal digits)

We can simplify Equation 2 as follows:

10x + y - 18 = 10y + x
9x - 9y = 18
x - y = 2 (Equation 3)

**Solving Equations 1 and 3:**

We have two equations with two variables. We can solve them simultaneously to find the values of x and y.

Adding Equation 1 and Equation 3, we get:

x + y + x - y = 10 + 2
2x = 12
x = 6

Substituting the value of x in Equation 1, we get:

6 + y = 10
y = 4

Therefore, the two-digit number is 64.

**Verification:**

Let's verify if the number 64 satisfies the given conditions.

The sum of the digits is 6 + 4 = 10.

When we subtract 18 from 64, we get 46. The digits in the resulting number (4 and 6) are equal.

Hence, the two-digit number that satisfies the given conditions is 64.