Assuming the spring obeys Hooke's law:
#F=kx#
Where
#F#=force
#k#=spring constant
#x#= extension
The question implies that #alpha# and #beta# is the total length of the spring, rather than the extension. If we define the length of the spring without any weights on it as #y#, then
We know that
#4=k(alpha-y)# .......equation (1)
and
#5=k(beta-y)# .......equation (2)
If we divide the two equations:
#5/4=(beta-y)/(alpha-y)#
We can expand and solve for y
#5(alpha-y)=4(beta-y)#
#5alpha-5y=4beta-4y#
#5alpha-4beta=y# ..........equation (3)
For the 9N weight, of total length, say #z#:
#9=k(z-y)#
If we divide this by equation (2)
#9/5=(z-y)/(beta-y)#
#9beta-9y=5z-5y#
#9beta-4y=5z#
Substitute in for y from equation (3):
#9beta-4(5alpha-4beta)=5z#
#9beta-20alpha+16beta=5z#
#25beta-20alpha=5z#
#z=5beta-4alpha#
