Name what we are looking for. 
Choose a variable to represent that quantity.

Let n = the first number. 
Let m = the second number.

Translate into a system of equations.

The sum of two numbers is 39. 

Their difference is 9. 

Translate into a system of equations.

The system is:

n + m = 39
n - m = 9

Solve the system of equations. 
To solve the system of equations, use elimination. 
The equations are in standard form and the coefficients of are opposites.

Add

n + m = 39
n - m = 9
_______________
2n = 48

Now solve for n

Substitute 24 for n into one of the original equations and solve for m.

n + m = 39

n - m = 9