Read the question to find out what to let the unknowns be:

The tens digit of the wife's age = t
The units digit of the husband's age = u
The wife's age is 10t+u

The tens digit of the husband's age = u
The units digit of the husband's age = t
The husband's age is 10u+t 

That tells us that the husband is older than the wife,
so we'll know which one to subtract from which one.
So difference in their ages = husband's age minus wife's age.

Difference in their ages = (10u+t)-(10t+u)

(10u+t)-(10t+u) = [(10u+t)+(10t+u)]

Simplify:

10u+t-10t-u = [10u+t+10t+u]

9u-9t = [11u+11t]

Multiply both sides of the equation by 11

11(9u-9t) = [11u+11t]

99u-99t = [11u+11t]

99u-99t = 11u+11t

88u = 110t

Divide both sides by 22

4u = 5t

 u = 

The left side u is a digit, so the right side must also be
that same digit.  The only digit t that will cause the
right side to be a digit is a multiple of 4.  8 or larger
won't do for 8 would make the right side 10 which is not
a digit.  So the only value for t is 4 itself. So t=4

Substituting:

 u = 

 u = 

 u = 5

So the wife is 45 and the husband is 54.

Edwin