12t + 64(8 - t) = 464

12t + 512 - 64t = 464

48 = 52t

Shouldn't it be

-48=-52t ?

It's same thing.

Either is fine. For the positive case, they added 52t to both sides to remove the one negative and 48 to both sides to remove the other negative.

Or multiplied both sides by -1.

Really it should be

12 = 13t

since you may as well divide through by the GCD. (And GCD(-48, -52) = GCD(48, 52) = 4). GCD = greatest common divisor. In algebra an equation that has all (non-zero) coeficients having a GCD of 1 (which 12 = 13t has, since 12 and 13 are coprime, which just means they have a GCD of 1) is called primitive. The content of a polynomial is the GCD of the coefficients (so a primitive polynomial has content 1).

EDIT: Whoops got my 12 and 13 mixed up there, lol.

EDIT2: It's similar to how fractions are expressed, you usually want an equation with coefficients that have a GCD of 1, they are in their "simplest" form (i.e. can't be trivially factored by dividing by the GCD of the coefficients to get another equation with all integer coefficients). You wouldn't give an answer involving fractions such as -48/-52, you would divide through by the GCD to get 12/13.

Thanks