The value of tension in a long thin metal wire has been changed from $$T_1$$ to $$T_2$$. The lengths of the metal wire at two different values of tension $$T_1$$ and $$T_2$$ are $$\ell_1$$ and $$\ell_2$$, respectively. The actual length of the metal wire is:

$$Y = \frac{\text{Stress}}{\text{Strain}} = \frac{T/A}{\Delta L/L} = \frac{TL}{A\Delta L}$$

$$\implies \Delta L = \frac{TL}{AY}$$

$$\ell_1 - L = \frac{T_1L}{AY} \implies \frac{L}{AY} = \frac{\ell_1 - L}{T_1} \quad \text{--- (1)}$$

$$\ell_2 - L = \frac{T_2L}{AY} \implies \frac{L}{AY} = \frac{\ell_2 - L}{T_2} \quad \text{--- (2)}$$

$$\frac{\ell_1 - L}{T_1} = \frac{\ell_2 - L}{T_2}$$ $$\implies T_2\ell_1 - T_2L = T_1\ell_2 - T_1L$$

$$\implies T_1L - T_2L = T_1\ell_2 - T_2\ell_1$$ $$\implies L(T_1 - T_2) = T_1\ell_2 - T_2\ell_1$$

$$\implies L = \frac{T_1\ell_2 - T_2\ell_1}{T_1 - T_2}$$