We are beginning with a 2-digit number. Let's call it "xy". If we reverse the digit we have "yx"
We know x+y=7
We also know that yx = 10x + 32
Using the first equation and solving for y, we have y=7-x
So in the 2nd equation the tens digit, y, is 7-x
Since the 7-x is the 10s digit, it's value is 10(7-x) and we can add this to the x in the yx
and we have
10(7-x)+x = 10x+32
If this is a bit unclear, think of it this way...
In the number 31, 3 times 10 + 1, 10 times the tens digit plus the one's digit.
Back to the problem
Rearrange terms to solve for x
Subtract 32 from both sides
38 - 9x = 10x
Add 9x to both sides
divide both sides by 19
x = 2
Since x+y=7, then y = 5
The original number is 25
The new number is 52 and 52 = 10(2) + 32