Why it works: Saṅkalana-vyavakalanābhyāṃ

If ax+by=p and bx+ay=q, then adding: $(a+b)(x+y)=p+q \to x+y=(p+q)/(a+b)$. Subtracting: $(a-b)(x-y)=p-q \to x-y=(p-q)/(a-b)$. Then x and y follow from $x=(sum+diff)/2$.