Two rods having length l₁ and l₂, and made of materials with the linear expansion coefficient α₁ and α₂, respectively, were soldered together. What is the equivalent coefficients of linear expansion for the newly obtained rod ?

Let the length of the combined rod system after expansion would be L. Now, from the question we have L = L₁ + L₂
Where L₁ be the length the length of one rod after expansion and L₂ be the length the length of another rod after expansion

$\begin{array}{l}{\mathrm{L}}_1={\mathrm{l}}_1\left(1+{\mathrm{\alpha }}_1\mathrm{\Delta T}\right),\\ L_2={\mathrm{l}}_2\left(1+{\mathrm{\alpha }}_2\mathrm{\Delta T}\right)\\ \mathrm{L}={\mathrm{l}}_1\left(1+{\mathrm{\alpha }}_1\mathrm{\Delta T}\right)\\ +{\mathrm{l}}_2\left(1+{\mathrm{\alpha }}_2\mathrm{\Delta T}\right)\\ \mathrm{L}={\mathrm{l}}_1+{\mathrm{l}}_2+\left({\mathrm{l}}_1{\mathrm{\alpha }}_1+{\mathrm{l}}_2{\mathrm{\alpha }}_2\right)\mathrm{\Delta T}\\ \mathrm{L}=\left({\mathrm{l}}_1+{\mathrm{l}}_2\right)\left(1+\frac{\left(l_1{\mathrm{\alpha }}_1+l_2{\mathrm{\alpha }}_2\right)\mathrm{\Delta T}}{l_1+l_2}\right)\\ \mathrm{\alpha }=\frac{l_1{\mathrm{\alpha }}_1+{\mathrm{l}}_2{\mathrm{\alpha }}_2}{{\mathrm{l}}_1+{\mathrm{l}}_2}\end{array}$