Biblical Contradictions - Murder

I have recently put forth a challenge via social media. I told my followers and friends to show me just one Biblical contradiction and I will stop following the Bible. The first one I received is referencing one of the 10 Commandments, Thou Shalt Not Kill (Exodus 20:13).
This seems very contradictory since God himself commands killing of certain peoples. However after a brief look into a concordance it becomes clear what is being said.
The Hebrew word for "Kill", or in some translations "Murder", in Exodus is 'ratsach.' This word has multiple meanings dependent on the context. If you are interested in studying this more in depth check this out.
Some of the meanings of this word are, slay or kill premeditated, accidental, as an avenger, as well as to murder, and assassinate.  The best way to to explain this in English, at least for me, is to say that murder is unjustified and killing is justified.
The Bible justifies killing in many places for various reasons.  For one example we can look at Exodus 22:2 "1If the athief is 2caught while breaking in and is struck so that he dies, there will be no bloodguiltiness on his account." This is clearly saying that if you happen to kill someone in defense of your self or property then it's justified. Another great example of this is Exodus 21:22-23 "If men strive, and hurt a woman with child, so that her fruit depart from her, and yet no mischief follow: he shall be surely punished, according as the woman's husband will lay upon him; and he shall pay as the judges determine. And if any mischief follows, then you shall give life for life..." This word mischief can be understood as the death of the unborn child. Some translations clarify it as such.

I hope this clears up any confusion to this apparent contradiction and if you have any questions feel free to comment! 

I'd have no guilt in my heart for killing someone who struck my wife and caused her to loose our unborn child. I know most wouldn't either, as in our heart this is a justified action. God allows this, and many other justified killings. Murder out of hate, rage, envy, or any other emotion is strictly forbidden though.

So in all the accounts where God has commanded someone kill, or where someone does kill in self-defense or defense of their family, it's justified and acceptable under the Law. 

God Bless!!