the one on the inside is just a forced one I think
and the one outside is natural not trying hard
if you have an algebraic equation you can modify it to change the form of the equation and still preserve the fact that the two sides are equal. the two operations that produce such modifications are
add the same number to each side of the equation and
multiply each side of the equation by the same number.
you want to attain the form ax + by = c, that is the x and y terms on the left side of the equation and a constant on the right side. you have 3y = 4x + 1 so you need to "move" the term 4x to the left side. this you can do by adding -4x to each side.
3y = 4x + 1, so
-4x + 3y = -4x + 4x + 1, which simplifies to
-4x + 3y = 1.
this is in standard form but it seems your son's textbook wants the coefficient of x to be positive not negative. this you can achieve by multiplying each side of the equation by -1.
-4x + 3y = 1, so
-1(-4x + 3y) = -1 × 1.
multiplying -1 through the left side gives
(-1) × (-4) x + (-1) × (3) y = -1, or
4x - 3y = -1.
i hope this helps