I am new to elliptic curve cryptography as well as finite field theory. I am trying to understand point addition in affine coordinates.
I understand, that for an elliptic curve $ y^{2}=x^{3}+ax+b $ over $\mathbb R$ the sum of two points $P=(x_{p},y_{p})$ and $Q=(x_{q},y_{q})$ is $R=(x_{r},y_{r})$: $$x_{r}=\lambda^{2}-x_{p}-x_{q}$$ $$y_{r}=\lambda(x_{p}-x_{r})-y_{p}$$
with the slope $$\lambda=\frac{y_{q}-y_{p}}{x_{q}-x_{p}}$$
Excluding the cases: $P=Q$ (e.g. tangent slope), $P=0$ and $Q=0$ (e.g. $R=0$). If however the elliptic curve is defined over a finite field with prime size $n$: $$y^{2}=x^{3}+ax+b\pmod n$$
Can I just compute the slope for the "standard case" as follows (source: Slide 6)?
$$\lambda=\frac{y_{q}-y_{p}}{x_{q}-x_{p}} \pmod n$$
I understand that for an element in a finite field (f.e. point $P=(x_{p},y_{p})$) amongst other things an multiplicative inverse has to exist. However the formula for the slope $\lambda$ does only include coordinates of the point, not the element itself.