Multiplying each side of the second equation by $\lambda$, we get:
$$\lambda (x' - ct') = \lambda × 0 =0 \space ,$$
Using this equation along with the first one, we easily get:
$$\lambda (x' - ct') = x - ct \space .$$
This seems to be saying $0 = \lambda × 0$ mathematically, which makes no sense.
Dandy! Why do you think it does not make sense?!
Second method:
The first and second equations imply:
$$x=ct, \space x'=ct'\rightarrow x=ct, \space \lambda x'=\lambda ct' \rightarrow$$$$x=ct, \space x'=ct'\rightarrow -x=-ct, \space \lambda x'=\lambda ct' \space.$$
Adding left sides and adding right sides gives:
$$\lambda x'-x=\lambda ct'-ct \rightarrow \lambda x'-\lambda ct'=x-ct \rightarrow$$
$$\lambda (x' - ct') = x - ct \space .$$