Say, given LDE $113 x +42y=1$ have solution given by $$113 = 2.42 +29\implies 29= 113 - 2.42$$ $$42= 1.29+13 \implies 13=42 -1.29$$ $$29= 2.13 +3 \implies 3=29 -2.13$$ $$13= 4.3 +1\implies 1=13 -4.3$$ $$3= 3.1 +0$$
Writing in reverse. $$13 -4.3=1$$ $$13 -4.(29 -2.13)=1\implies -4.29 + 9.13= 1$$ $$-4.29 + 9.(42-29)=1\implies -13.29 + 9.42= 1$$ $$9.42 -13.(113-2.42)= 1\implies 35.42-13.113= 1$$
So, one solution is $(X,Y)= (-13, 35)$.
Learned that for LDE $113 x -42y=1$, one solution $(X,Y)= (-13, -35)$
How it is obtained is unclear.
Say, applying the same process to LDE $113 x -42y=1$ get:
$$113 = (-2).(-42) +29\implies 29= 113 + 2.(-42)$$ $$(-42)= (-1).29+(-13) \implies -13=(-42) + 1.29$$ $$(-29)= 2.(-13) -3 \implies -3=(-29) -2.(-13)$$ $$(-13)= 4.(-3) -1\implies 1=(13)+4.(-3)$$ $$-3= -3.1 +0$$
Writing in reverse. $$13 +4.(-3)=1$$ $$13 +4.(-29 -2.(-13))=1\implies -4.29 + 9.13= 1$$ $$-4.29 + 9.(42-29)=1\implies -13.29 + 9.42= 1$$ $$9.42 -13.(113-2.42)= 1\implies 35.42-13.113=1$$
So, where erred in not getting $-35.42-13.113=1$
Cannot understand if this approach is flawed of getting solution for opposite sign of $b$, i.e. need an alternative way; or erred above.
If alternative approach is the only way; say, sign change of $b$ means simply sign change of $Y$; then need a better way to understand it.
I think your approach is fine, but from $35\cdot 42 - 13\cdot 113 = 1$, you get $113\cdot (-13)-42\cdot(-35)=1$ for a solution of $(-13,-35)$, the expected result.
But simpler is to observe: $$\begin{array}{ccl} (a,b) {\rm\;is\; a\; solution\; of\;} 113x+42y &\Rightarrow& 113a+42b=1\\ &\Rightarrow&113a-42\cdot(-b) =1 \\ &\Rightarrow& (a,-b) {\rm \;is\; a\; solution\; of\;} 113x-42y=1 \end{array}$$