Jenny's favorite number is a two-digit positive integer. if she reverses the digits, the new number is 27 less than her favorite number. if she instead halves the tens digit and triples the units digit of her favorite number, the new number she forms is 18 less than her favorite number. what is jenny's favorite number

Suppose her favorite number is xy.
where x is the tens place digit and y is the units place digit.
xy can also be written as = 10x + y

The reversed number will be yx, which can also be written as = 10y + x
Reversed number  is 27 less than the original favorite number, so we can write the equation as:
10y + x + 27 = 10x + y 
Simplifying the equation, we get:
9x - 9y = 27
9 ( x- y ) = 27
x - y = 3
x = 3 + y

If Tens digit number if halved, and unit digit is tripled the new number will be = 10(x/2) + 3y = 5x +3y
This number is 18 less than her original favorite number.

So we can write the equation as:
5x + 3y + 18 = 10x + y
Simplifying, we get:
5x - 2y = 18  
Using the value of x = 3 +y in above equation, we get:

5(3+y) - 2y = 18
15 + 5y - 2y = 18
3y = 3
y = 1

So, x = 3 + y = 3 + 1 = 4

Thus her favorite number is xy = 41