Algebra

By far, model diagrams will be the easiest for visualisation. Algebraic approaches are not very suitable at your level. There are alternative approaches too, but very hard to visualise without a model diagram to help you.

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If Teddy gave Lynn 150 seashells, Teddy woould have 150 fewer seashells (than before) and Lynn would have 150 more seashells (than before). Teddy would have the same number of seashells than Lynn in this scenario.

This means that originally, Teddy must have had 150 more seashells than the "both would have the same number of seashells" scenario, while Lynn must have had 150 fewer seashells than the "both would have had the same number of seashells" scenario. As a result, originally Teddy must have had 300 more seashells than Lynn.

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If Lynn gave Teddy 150 seashells, Lynn would have 150 fewer seashells than the original scenario and Teddy would have 150 more seashells than the original scenario. The gap between Teddy and Lynn would widen by a further 300 seashells, making Teddy have 600 seashells more than Lynn in this scenario.

Coincidentally, in this scenario, Teddy would have 3 times as many seashells as Lynn.

This means that Teddy would have an extra 2 units of seashells as compared to Lynn, which would corresponds to the extra 600 seashells.

2 units = 600 seashells
1 unit = 300 seashells

So in this scenario (after Lynn gave away the 150 seashells), Lynn would have 300 seashells.

Number of seashells Lynn had at first
= 300 + 150
= 450

(and if you make further calculations, Teddy had 750 seashells at first).