When the tension in a metal wire is ${\mathrm{T}}_1$, its length is ${\mathrm{l}}_1$. When the tension is ${\mathrm{T}}_2$, its length is ${\mathrm{l}}_2$.The natural length of wire is

$\mathrm{Y} = \frac{\mathrm{Fl}}{\mathrm{a}∆\mathrm{l}}$

Y, l and a are constants

$\therefore \frac{\mathrm{Fl}}{∆\mathrm{l}} = \mathrm{constant} \mathrm{or} ∆\mathrm{l} \propto \mathrm{F}$

Now, ${\mathrm{l}}_1-\mathrm{l} \propto {\mathrm{T}}_1 \mathrm{and} {\mathrm{l}}_2-{\mathrm{l}}_1\propto {\mathrm{T}}_2$

Dividing, $\frac{{\mathrm{l}}_2-\mathrm{l}}{{\mathrm{l}}_2-\mathrm{l}} = \frac{{\mathrm{T}}_1}{{\mathrm{T}}_2}$

or ${\mathrm{l}}_1{\mathrm{T}}_2-{\mathrm{lT}}_2 = {\mathrm{l}}_2{\mathrm{T}}_1-{\mathrm{lT}}_1$

or $\mathrm{l}({\mathrm{T}}_1-{\mathrm{T}}_2) = {\mathrm{l}}_2{\mathrm{T}}_1-{\mathrm{l}}_1{\mathrm{T}}_2$

or $\mathrm{l} = \frac{{\mathrm{l}}_2{\mathrm{T}}_1-{\mathrm{l}}_1{\mathrm{T}}_2}{{\mathrm{T}}_1-{\mathrm{T}}_2} \mathrm{or} \mathrm{l} = \frac{{\mathrm{l}}_1{\mathrm{T}}_2-{\mathrm{l}}_2{\mathrm{T}}_1}{{\mathrm{T}}_1-{\mathrm{T}}_2}$