3. Britain was the first country in the world to have a free health care system paid for by the government.
英國是世界上第一個擁有由政府負擔費用的免費醫療保健系統的國家。
(1)這個句子的重點是the first...to do “第一個干……的”,動詞不定式作定語,相當于一個定語從句。句中to have 可以轉換為that / which has。
As we all know, Wang Zhizhi is the first player in Asia to join the NBA.
眾所周知，王治郅是到NBA打球的亞洲第一人。
He was the last to be there.
他是最后一個到那兒的。
(2) paid for by the government 是過去分詞短語作后置定語,相當于一個定語從句which was paid for by the government,修飾先行詞system。
The student dressed in white is my daughter.
=The student who is dressed in white is my daughter.
穿白色衣服的學生是我的女兒
(3) pay for 意為“付款”，“付出代價”。
He was in such a hurry that he left quickly without paying for the vegetables.
他太匆忙了，沒付買蔬菜的錢就匆匆離開了。
You can't be too careful, or you'll pay for it.
你越小心越好,否則你會為此付出代價的。

4. The problem with this system is that poorer people don't have the money to pay for private health insurance.
這種制度所存在的問題是窮人沒有錢繳納個人健康保險。
該句是復合句。that在此引導表語從句,在句中無意義，但不能省略。that除了引導表語從句之外，還可以引導主語從句、同位語從句和賓語從句,在賓語從句中可以省略，其他從句中都不可以省略。
The fact is that he has not been seen recently. (that引導表語從句)
事實是近來誰也沒有見過他。
John said that he was leaving for London on Wednesday.(that引導賓語從句，可省)
約翰說他星期三要到倫敦去。
The fact that he has not been seen recently disturbs everyone in his office.(that引導同位語從句)
近來誰也沒有見過他,這一情況令辦公室所有的人不安。
It is quite clear that the whole project is doomed to failure.(that引導主語從句)
很清楚，整個計劃注定要失敗。