Your equation (2) is wrong because you use Newton's 1st law in the tilted direction.

In that direction there is a component of acceleration, so Newton's 2nd law should have been used, not the 1st.

The point is that in this tilted direction (perpendicular to the slope), the horizontal acceleration has a parallel component (a component perpendicular to the slope). So this acceleration component must be included in any Newton's laws you set up - meaning, you now must use Newton's 2nd law $\sum F=ma$ rather than Newton's 1st law $\sum F=0$ since there is an acceleration component present.

Since we are more used to work with *force components* than with *acceleration components*, you will often see teachers and answer sheets pick a coordinate system that fits the acceleration - and thus *not* necessarily one which is tilted along with the slope. In that way they avoid acceleration components and only have full acceleration along one axis and none along the other.