Let $G$ be a graph. If $M$ and $N$ are two disjoint matchings $(M\cap N = \emptyset)$ in $G$ with $|M| > |N|$, then there are disjoint matchings $M_1$ and $N_1$ such that $|M_1| = |M| - 1$, $|N_1| = |N| + 1$, and $M_1\cup N_1 = M\cup N$.
I'm having trouble with this problem because while I understand that we're shifting edges between the two matchings, I have no idea how to prove that can be done.
Think about the union $M \cup N$ of the two matchings.
This is a set of edges forming a subgraph of $G$ in which every vertex has maximum degree $2$. More precisely, every vertex is incident to at most one edge from each matching. You might imagine coloring the edges of $M$ red and the edges of $N$ blue; every vertex is incident to a red edge, or a blue edge, or neither, or both.
So what can the connected components of this subgraph look like? (First think about what kinds of connected graphs have maximum degree $2$, and then about what the colors on the edges will be for each kind.)
What kind of component can you use to increase the size of $N$ (and decrease the size of $M$)? And how can you argue that such a component must exist?