Let

x = first number

y = second number

First number plus twice a second number is 12:

x + 2y = 12

Twice the first number plus the second totals 18:

2x + y = 18

by solving the system of equations:

x + 2y = 12

2x + y = 18

we find:

x = 8

y = 2

click here to see the step by step solution of the system of equations

Indeed:

8 + 2*2 = 12

2*8 + 2 = 18

The numbers are 8 and 8.