Haha, don't worry, you're not alone.
I'm 14 years old, amd 5 feet and 2 inches, and I weigh 85 pounds. Even though I love to go outside and play tag, I never get to because my brother won't, and I can't do anything without supervision. >,<
So I'm usually on CS, I eat normal (except for one night when I was really stressed, I skipped dinner), and even though I want to, I don't get enough excersise. And I can't gain weight.
But don't worry, just because you're skinny doesn't mean you're weak. I'm not very strong when it comes to fights, but I can do very well in gym class. I tied for the most push-ups in the girl's division last year in seventh grade with an eighth grader (36 push-ups, and you had to touch the teacher's hand that was flat on the ground with your chest), and I came in first in the girls division in the mile run. Yet I still don't have muscle. x3
But, I'm pretty sure if you work out, you'll add on some muscle and weigh more. There's tons of things you can do to get muscle: bench-press, push-ups, pull-ups, sit-ups... Tons of things. Some schools have after-school trips to a local gym, or you could just improvise and use house-hold items to lift, use as your puch-up marker, etc. Even though you're excersising, you could still gain weight from putting on muscle.
Hope this helped!