x+2y=10 so -2x-4y=-20 (by multiplying through by -2).
If we add the second equation: -4y+3y=-20+14, so -y=-6, that is, y=6 (multiply through by -1).
Now substitute y=6 in either original equation. x+12=10, x=10-12=-2; 2x+18=14, 2x=14-18=-4, x=-2 (divide through by 2). Solution: x=-2, y=6.
Another way: Double the first equation and subtract the second from the result:
2x+4y=20 minus
2x+3y=14
y=6 (as before) then substitute for y in either equation, as before, to get x.
There are other ways to eliminate one variable to reduce the problem to a single equation containing only one variable, which can be solved.