Let "a" be the tens digit of the number and "b" be its ones digit.
So the number itself is 10a + b.

You are given

10a + b = 4(a+b) + 3       (1)    ("A two digit number is 3 more than 4 time the sum of its digits")
(10a + b) + 18 = 10b + a   (2)    ("If 18 is added to the number, the digits are reversed.   Notice:  10b + a  is  10a + b after reversing digits !)

So, you have this system of two equations (1) and (2) for the two unknowns "a" and "b".

Let us rewrite the system in the standard form:

6a - 3b =   3,     (1')
9a - 9b = -18.     (2')

Cancel common factor 3 in (1') (in both sides)  and  9  in (2') (both sides). You will get

2a - b =  1,       (1'')
 a - b = -2.       (2'')

Now distract equation (2'') from equation (1''). You will get

a = 3.

Then from (2'')  b = a+2 = 3+2 = 5.

Answer.  Your number is 35.

Check.   Check it on your own.