$解：由题意可得\ \left\{\begin{array}{l}2 x+5 y=-6 \\ 3 x-5 y=16\end{array}\right. ,解得:\ \left\{\begin{array}{l}x=2 \\ y=-2\end{array}\right. ,$
$将\ \left\{\begin{array}{l}x=2 \\ y=-2\end{array}\right.\ 分别代入a x-b y=-4 ,b x+a y=-8 \text { 中得 }\left\{\begin{array}{l}2a+2b=-4 \\2b-2a=-8\end{array}\right. \text {, }$
$解得:\ \left\{\begin{array}{l}a=1 \\ b=-3\end{array}\right. ,$
$则(2a+b)^{2024}=(2 \times 1-3)^{2024} =(-1)^{2024}=1.$
