Solve for y:
$$6y-3-(2y-1)=12\tag1$$
$$6y-3-2y-1=12\tag2$$
$$4y-4=12\tag3$$
$$4y=12+4\tag4$$
$$y=4\tag5$$
But when I plugged into $(1)$ it is wrong,
$y=4$ $$6y-3-(2y-1)=12\tag1$$
$$24-3-(8-1)=12\tag a$$
$$24-3-(7)=12\tag b$$
$$14=12\tag c$$
Where it is wrong?
First of all, congratulations on realizing there must be a mistake somewhere. Others have pointed out what's wrong, so that by now it may be obvious. But for future reference, you can sometimes localize the step at which the error occurred by plugging the (incorrect) answer into some of the other equations as well; if it satisfies an intermediate equation, then you know the error occurred earlier.
In this case, $y=4$ satisfies equation (2):
$$6y-3-2y-1=12\\ 24-3-8-1=12\\ 21-8-1=12\\ 13-1=12\\ 12=12$$
This means that the error occurred in going from (1) to (2). But all you did there was to remove parentheses, which seems innocuous enough. How could anything go wrong? Ah! That minus sign in front of the parentheses! You have to distribute it!
Another thing to do is to take note of mistakes that you tend to make on a regular basis, and slow down when you recognize you're about to take a step that you know you've often done wrong. For you it might be minus signs; for me it's pointing inequality signs in the proper direction (among many other idiosyncratic mistakes). Mathematics doesn't have to be done in a hurry.
To ensure this has an answer, look at your leap from (1) to (2). The term $-(2y-1)$ should have been expanded as $-2y+1.$ Thus, (2) should be $$6y - 3 -2y +1 = 12.$$
I assume you can take it from here.
Note also that this is an excellent way to do math: left to right, top down. You should be able to retrace your steps and determine if you've made any mistakes. You should go through this with a fine comb, as the mistake may not instantly pop out.
