Suppose two numbers are x and y. By the statement of the problem we will get the following

x+y=34 eq 1from here x=34-y

put in the second equation

x-y=10 eq 2

34-y-y=10

34-2y=10

2y=34-10

2y=24

y=12

put in eq 1

x+12=34

x=34-12

x=22 so the two numbers are 22 and 12

x+y=34 eq 1from here x=34-y

put in the second equation

x-y=10 eq 2

34-y-y=10

34-2y=10

2y=34-10

2y=24

y=12

put in eq 1

x+12=34

x=34-12

x=22 so the two numbers are 22 and 12