"she said the best way to learn new words was by reading English magazines"中为什么要用“was”而不是直接省略呢?还有就是 I study by making flashcards中就没有was?

she said that
study

was====be V.
Study=== V.
S + V + O............

she said the best way to learn new words was by reading English magazines
这是一个含宾语从句的复合句,主句是she said
the best way to learn new words was by reading English magazines.是宾语从句,既然是句子,英语中就得要有主语谓语,the best way to learn new words 是从句的主语,was by reading English magazines
构成谓语部分
而I study by making flashcards.是简单句,简单句的定义是：只有一个主语（或并列）和一个谓语（或并列）的句子叫简单句.
这位同学你题的问题挺好,要学好英语就的一点一点的学习,直至学会弄懂

i study by 翻译过来就是我学习用……什么什么的，这里的什么指的是一样东西，或者一个动词，不需要is, was来关联。
she said the best way to learn new words was...翻译就是她说最好的学习方法是（was)，这里was就是这个‘是’，就算是中文也要加进去，英文也是一样。
两句话的意思不同，所以结构自然不同。