First, let's define the two numbers we are looking for. I will call them:

#n# and #m#.

Next, from the problem we know we can write:

#n + m = 19# and #n - m = 7#

Then, solve the second equation for #n#;

#n - m = 7#

#n - m + color(red)(m) = 7 + color(red)(m)#

#n - 0 = 7 + m#

#n = 7 + m#

Now, substitute #7 + m# for #n# in the first equation and solve for #m#:

#n + m = 19# becomes:

#7 + m + m = 19#

#7 + 2m = 19#

#-color(red)(7) + 7 + 2m = -color(red)(7) + 19#

#0 + 2m = 12#

#2m = 12#

#(2m)/color(red)(2) = 12/color(red)(2)#

#(color(red)(cancel(color(black)(2)))m)/cancel(color(red)(2)) = 6#

#m = 6#

We have found the first number. To find the second number substitute #6# for #m# back into the equation we solved for #m# and calculate #n#:

#n = 7 + m# becomes:

#n = 7 + 6#

#n = 13#

The two numbers are #6# and #13#

#6 + 13 = 19# and #13 - 6 = 7#