The length of a metal is l₁ when the tension in it is T₁ and is l₂ when the tension is T₂. Find the original length of the wire.

Let the original length of the wire be 'L'
now, we have

in the first case
L₁ = L + dl₁
or [from Hooke's Law;  x = T/k]

L₁ = L + (T₁/k)   .................(1)

similarly, for the second case we will have
L₁ = L + dl₂
or
L₂ = L + (T₂/k)  .................(2)

we can now use (1) and (2) to solve for original length 'L'
so, we get

L = (L₁T₂ - L₂T₁) / (T₂ - T₁)