Question

why are both the x-coordinate and y-coordinate generally needed to find the slope of the tangent line at a point for an implicitly defined function

Answer 1

The tangent slope is *defined* as $\frac{\Delta y}{\Delta x}=\frac{y_1-y_0}{x_1-x_0}$. (this relates to the function tangens, tan, of an angle).

Because it is defined as a ratio of the change in y to the change in x, both coordinates are needed. Having only one coordinate, the slope would be undetermined. Lmk if you have questions.