5、I sincerely hope that you can give us a favorable reply at your early convenience.

【译文】我真诚希望您能尽早在方便之时给我一个肯定 的答复。

【分析】本句常用于邀请信的结尾处，是比较正式的书面用语，表达真诚希望对方接受邀请之意。favorable 在本句中意为“赞同的，同意的”。at one's convenience “在某人方便之时”。 a favorable reply at your early convenience 可以替换为 an early and favorable reply。

6、According to the picture, we can safely come to a conclusion that our life is a course that requires constant pursuit of greater goals.

【译文】根据图画,我们可以确切地得出结论:生命是一个需要不断追求更大目标的过程。

【分析】本句常用于具体描述图中内容后给出结论，然后就此展开论述。According to the picture可以替换为Based on the picture，come to a conclusion 可以替换为 draw/reach/make a conclusion，也可以直接用动词咖conclude，或直接用 It's clear that，clearly 等。