I have two quantities $a$ and $b$ which satisfy $a+b=13$. I want to find the values of $a$ and $b$ for which, they are in the golden ratio, i.e. $\frac{a}{b}=\frac{1}{2} \left(\sqrt{5}+1\right)$. My question is :
Should I solve $a+b=13$ and $\frac{a}{b}=\frac{1}{2} \left(\sqrt{5}+1\right)$ to find the values?
or I should solve $\frac{a}{b}=\frac{1}{2} \left(\sqrt{5}+1\right)$ and $\frac{a+b}{a}=\frac{13}{a}=\frac{1}{2} \left(\sqrt{5}+1\right)$?
As usual let $\varphi=\frac12\left(1+\sqrt5\right)$ be the golden ratio. You know (or can easily check) that $\varphi^2=\varphi+1$, so if $a=b\varphi$, then $$a\varphi=b\varphi^2=b\varphi+b=a+b=13\,.$$ Clearly the system
$$\left\{\begin{align*} &b\varphi=a\\ &a\varphi=13 \end{align*}\right.$$
is a very easy one to solve, first for $a$ and then for $b$.