What Velocity is Needed for a Rocket Pilot to Reach a Star in One Year?

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To determine the velocity required for a rocket pilot to reach a star 4 light years away in one year of the pilot's time, relativistic effects must be considered. The Lorentz factor plays a crucial role in relating the proper time and distance from the pilot's perspective to the stationary observer on Earth. The discussion highlights confusion regarding a specific step in the provided solution, particularly the transition from lines 6 to 7. A substitution from relevant formulas in special relativity is suggested, but the user is uncertain about the correct application. Clarification on these calculations is sought to better understand the problem.
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Homework Statement



The pilot of a rocket wants to reach a star in 1 year. The rocket is currently on Earth. The star is 4 light years away, measured from an observer on Earth. What velocity is needed for the pilot to reach the star in a year in the pilot's/rocket's time?

Homework Equations



See below. Relativistic Length = Proper Length / Lorentz factor is used below, to make the substitution from Line 2 to Line 3 of L to L'/(Lorentz factor). The gamma symbol indicates the Lorentz factor.

The Attempt at a Solution



I have been given a solution, I just don't understand it. Is someone able to explain the jump from lines 6 to 7 in this solution? http://i.imgur.com/F70Oy.jpg . T' and L' indicate the proper time (time is from pilot's perspective, hence 1 year, distance from Earth's perspective, hence 4 light years). Tp is used the in first line, it is the same as T', hence where the T' comes from in Line 4.

I have been struggling with this for some time. Any help is appreciated, thank you!

Edit: I believe there is a substitution of a formula from this section of Wikipedia but I am unsure which one. Also, Wikipedia uses different symbols (in each formula, variables without a dash indicate measurements made in the stationary reference frame, unlike mine). https://en.wikipedia.org/wiki/Special_relativity#Time_dilation_and_length_contraction
 
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Why not just do it yourself? I'll get you started

t'=d'/v
 

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