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3y+53=7y-55       add 55 to BOTH sides   and subtract 3y from BOTH sides to get:

108 = 4y               divide both sides by 4 to get

27 = y

$3y+53=7y-55$

$3y=7y-108$

$-4y=-108$

$y=27$

Replace $y$ with 27 in the original problem to check if the answer is correct.

$3\times27+53=7\times27-55$

$81+53=7\times27-55$

$134=7\times27-55$

$134=189-55$

$134=134$

That checks out.  The answer of $y=27$ is correct.