Two different wires having lengths L _ 1 and L _ 2 and respective...

The wire are joined end-to-end, so they are in series combination, then total length of wire is given by
$L = {L}_{1} + {L}_{2} . . . ( 1 )$

When the temperature is increase by $∆ \theta$, then its final length is given by using concept of linear expansion,
${L}^{'} = L ( 1 + \alpha ∆ \theta ) . . . ( 2 )$

Using equation $( 1 )$ and $( 2 )$ for the series combination of wires, final length is given by

${L}^{'} = {{L}_{1}}^{'} + {{L}_{2}}^{'} . . . ( 3 )$

$⇒ \left({L}_{1} + {L}_{2}\right) ( 1 + {\alpha}_{eff} ∆ \theta ) = {L}_{1} ( 1 + {\alpha}_{1} ∆ \theta ) + {L}_{2} ( 1 + {\alpha}_{2} ∆ \theta )$

$⇒ {\alpha}_{eff} = \dfrac{{L}_{1}{\alpha}_{1} + {L}_{2}{\alpha}_{2}}{\left({L}_{1} + {L}_{2}\right)}$