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Newton's second law states that net force equals mass times acceleration, so accelerating 12 kg at 3.5 m/s2 needs 42 N. The weight is shown alongside because it is the force gravity already applies - 117.7 N here - and any real push must overcome whatever fraction of it acts along the motion.
Newton's second law
F = m x a
Weight
W = m x g
It is the net force, the vector sum of everything acting. If friction opposes you, your push must be larger by the friction force.
Because inertia depends on mass, which does not change with location. Weight would give the wrong answer on the Moon.