Construction of Elliptic Curve with given $j$-invariant

Question

Recall the following well-known result. Let $K$ be a number field, and let $j_0 \in \overline{K} \setminus \{1728\}$. Then there is an elliptic curve defined over the field $K(j_0)$ with $j$-invariant equal to $j_0$, and in the affine plane, one such curve can be thought of as being cut out by the Weierstrass equation $$y^2 + xy = x^3 - \frac{36}{j_0 - 1728} x - \frac{1}{j_0 - 1728}.$$ It is easy to check that the $j$-invariant associated to the above Weierstrass equation is in fact $j_0$, but how does one actually derive this formula without knowing the answer? Or did mathematicians just stumble on it?