Let's use algebra to solve this problem. Let the two-digit number be represented as 10a + b, where 'a' is the tens digit and 'b' is the units digit.

We are given two conditions:

The sum of the digits is 10, so we can write the equation: a + b = 10

When 72 is subtracted from the number, the digits are interchanged. This means that 10a + b - 72 becomes 10b + a. We can write this as another equation: 10a + b - 72 = 10b + a

Now, let's solve this system of equations:

From equation 1, we can express 'b' in terms of 'a': b = 10 - a

Substitute this expression for 'b' into equation 2: 10a + (10 - a) - 72 = 10(10 - a) + a

Now, simplify and solve for 'a': 10a + 10 - a - 72 = 100 - 10a + a

Combine like terms: 9a + 10 - 72 = 100 - 9a

Subtract 10 from both sides: 9a - 62 = 100 - 9a

Add 9a to both sides: 18a - 62 = 100

Add 62 to both sides: 18a = 162

Divide by 18: a = 9

Now that we have found 'a' to be 9, we can find 'b' using the first equation: a + b = 10 9 + b = 10

Subtract 9 from both sides: b = 10 - 9 b = 1

So, the tens digit 'a' is 9, and the units digit 'b' is 1. Therefore, the two-digit number is 91.