Weed and drinking made my life hell

How can I make my mom trust me after I told her I didnt drink or smoke weed then she found out?

Its going to take time, your going to have to earn her trust all over again. You cant force it to happen any quicker.

Sounds like you have an uphill battle going. . .
I suggest for starters you quit drinking and Smoking.
Also both are illegal at your age so they can both get you in bigger trouble than your in with your mom.

make yourself seem more responsible. tell her you made a mistake, and you will try hard to make up for the lies. this will most likely work

You have got to come up with ways to rebuild trust and show her you can be reponsible - first, I'd apologize for my poor choices and behavour, seek help, and go from there...You can do it !!

admit you made a mistake...
it will take a while...

 
You have to learn to trust her and her decisions. Earning her trust will take time but it can be done. You might also (assuming that you have already apologized to her) try asking her what some of the things are that you can do to earn back her trust.
 

Theres a saying that goes like this: 'Responsibility to the responsible, not to those who want free meals'

Show her that you're responsible, and just made a mistake. Its a simple matter of doing the things that you're supposed to do.
At least, its simple on paper. But if you truely sorry, then your energy will be spent helping regain your trust.