Can he slaughter the udhiyah with his left hand?

Praise
be to Allah

The scholars regard it as mustahabb to slaughter with the
right hand, because that is easier for the animal, as most people use
the right hand more, so it is stronger. 

As for slaughtering with the left hand, it is contrary to
what is preferable, unless slaughtering with the left hand is easier for the
person and for the animal, in which case he may slaughter with the left
hand, and that is preferable in his case to slaughtering with the right
hand. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:

It is not essential that slaughter be done with the right
hand; rather it is permissible to do it with the right hand or the left
hand, because the Prophet (blessings and peace of Allah be upon him) said:
“If blood has been caused to flow and the name of Allah has been pronounced
over it, then eat.” And he did not stipulate that that should be done with
the right hand, but undoubtedly doing it with the right hand is better,
because it is stronger, and if it is stronger then it will be easier for the
animal. The Prophet (blessings and peace of Allah be upon him) enjoined
making it easier for the animal when he (blessings and peace of Allah be
upon him) said: “Allah has decreed competence in all things. So when you
kill, kill well and when you slaughter, slaughter well, and let one of you
sharpen his blade and spare suffering to the animal he is going to kill.” 

Based on that, slaughtering with the left hand may be
preferable to slaughtering with the right hand, such as if a person is
left-handed and he uses his left hand and not his right hand. In this case,
it is better to slaughter with the left hand, because it is stronger, and
that will be easier for the animal. 

To sum up our answer: we say that there is nothing wrong with
a person slaughtering with his left hand, because the Prophet (blessings and
peace of Allah be upon him) did not stipulate that slaughter should be done
with the right hand. 

See Fataawa Noor ‘ala ad-Darb on the following link: 

http://www.ibnothaimeen.com/all/noor/article_1871.shtml 

A similar fatwa was issued by His Excellency Shaykh Ibn Baaz
(may Allah have mercy on him), as you will find on the following link: 

http://www.binbaz.org.sa/mat/19839 

See also the answer to question no.
162299.