Problem Analysis:
Let's assume the original two-digit number as "ab", where "a" represents the tens digit and "b" represents the ones digit.

Given that the sum of the digits is 7, we can write the equation as:
a + b = 7 ...(1)

If the digits are reversed, the new number becomes "ba". According to the problem, this new number increased by 3 equals 4 times the original number. Mathematically, we can express this as:
10b + a + 3 = 4(10a + b) ...(2)

Solving the Equations:
To find the original number, we need to solve the equations (1) and (2) simultaneously.

Simplifying Equation (2):
Expanding the right side of equation (2), we get:
10b + a + 3 = 40a + 4b

Rearranging the terms, we obtain:
36a - 6b = -3 ...(3)

Solving Equations (1) and (3) simultaneously:
We can solve equations (1) and (3) by substitution or elimination method.

Using the Substitution Method:
From equation (1), we can express "a" in terms of "b" as:
a = 7 - b

Substituting this value of "a" in equation (3), we get:
36(7 - b) - 6b = -3

Expanding and simplifying the equation:
252 - 36b - 6b = -3
-42b + 252 = -3
-42b = -255
b = 6

Substituting the value of "b" back into equation (1), we find:
a + 6 = 7
a = 1

Therefore, the original two-digit number is 16.

Verification:
Let's verify whether our solution is correct by checking if both conditions are satisfied:

1. Sum of the digits is 7: 1 + 6 = 7 (satisfied).
2. Reversed number increased by 3 equals 4 times the original number:
Reversed number = 61
61 + 3 = 4 * 16
64 = 64 (satisfied).

Hence, our solution is correct. The original two-digit number is 16.