back 以前；过去；追溯至( 英文解释：If you talk about something that happened back in the past or several years back, you are emphasizing that it happened quite a long time ago.)

The story starts back in 1950, when I was five.

故事可以追溯到1950年，那时我才5岁。

I was in St. Lucia back in January of this year.

今年1月份我在圣卢西亚。

Excuse me,I received a letter that I am supposed to return a book that I checked out back in September

以上副词back属于以上用法，而且后面经常带有介词in.

check out用法如下：

(1)check out结账退房;结账离开(诊所);给…办理结账手续 (英文解释：When you check out of a hotel or clinic where you have been staying, or if someone checks you out, you pay the bill and leave.)

They packed and checked out of the hotel. 

他们收拾好东西，办理了退房手续。

I was disappointed to miss Bryan, who had just checked out. 

我非常失望没有见着布赖恩，他刚退房走了。

(2)check out 了解清楚;核实;查实 (英文解释：If you check out something or someone, you find out information about them to make sure that everything is correct or satisfactory.)

Maybe we ought to go down to the library and check it out. 

或许我们应该去趟图书馆，查个明白。

We ought to check him out on the computer.  

我们理应在电脑上查清他的资料。

(3)check out经证实是对的;符合要求 (英文解释：If something checks out, it is correct or satisfactory.)

She was in San Diego the weekend Jensen got killed. It checked out. 

詹森遇害的那个周末她在圣迭戈，后经查证情况属实。

Everything checks out.  

查过了，一切正常。