The length of a metal wire is $$\ell_1$$, when the tension in it is $$T_1$$ and is $$\ell_2$$ when the tension is $$T_2$$. The natural length of the wire is:

Let $$\ell_0$$ be the natural length and $$Y$$ be Young's modulus, $$A$$ the cross-sectional area. By Hooke's law, extension $$= \frac{T \ell_0}{YA}$$, so the stretched length is $$\ell = \ell_0 + \frac{T \ell_0}{YA} = \ell_0\left(1 + \frac{T}{YA}\right)$$.

For the two cases: $$\ell_1 = \ell_0\left(1 + \frac{T_1}{YA}\right)$$ and $$\ell_2 = \ell_0\left(1 + \frac{T_2}{YA}\right)$$.

Let $$k = \frac{\ell_0}{YA}$$, so $$\ell_1 = \ell_0 + kT_1$$ and $$\ell_2 = \ell_0 + kT_2$$. From these two equations: $$\ell_2 - \ell_1 = k(T_2 - T_1)$$, giving $$k = \frac{\ell_2 - \ell_1}{T_2 - T_1}$$.

Substituting back: $$\ell_1 = \ell_0 + \frac{(\ell_2 - \ell_1)T_1}{T_2 - T_1}$$, so $$\ell_0 = \ell_1 - \frac{(\ell_2 - \ell_1)T_1}{T_2 - T_1} = \frac{\ell_1(T_2 - T_1) - T_1(\ell_2 - \ell_1)}{T_2 - T_1} = \frac{\ell_1 T_2 - \ell_2 T_1}{T_2 - T_1}$$.

Therefore the natural length of the wire is $$\ell_0 = \dfrac{\ell_1 T_2 - \ell_2 T_1}{T_2 - T_1}$$.