<p>We investigate the solvability of two dual quaternion matrix equation systems <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3342_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="160" /> </InlineMediaObject> <EquationSource Format="TEX">\((AX, CXD) = (B, E)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mi>X</mi> <mo>,</mo> <mi>C</mi> <mi>X</mi> <mi>D</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40314_2025_3342_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="161" /> </InlineMediaObject> <EquationSource Format="TEX">\((XA, DXC) = (B, E)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mi>A</mi> <mo>,</mo> <mi>D</mi> <mi>X</mi> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> via generalized inverses and matrix rank, establishing necessary and sufficient conditions for consistency and providing the general solution in solvable cases. These results are extended to several special cases. To support our findings, we conduct a numerical experiment that validates the results. Additionally, we present an application of color image encryption and decryption based on the derived solutions.</p>