Can a person with HIV live long?

Yes, if given appropriate treatment promptly. If the right drugs are given, the viral load in the body can be kept under control for a long time; some people are now not developing AIDS for 20 years or more after their diagnosis!
However, this doesn't mean it's a disease to be taken lightly. The drugs used can have some nasty side-effects, and it's not pleasant to have to take a load of pills, three times a day, for life. And if the drug regime isn't strictly adhered to, it won't be effective. And, of course, the person can always infect others with HIV, even if they don't feel unwell.
But the life expectancy of an HIV-positive person is far higher now than it used to be.

These days, with treatment, people with HIV can live longer, healthier than they could before. People who follow their treatment plan as prescribed can live for many years. (However, the length of time a person lives would depend on the person's health prior to infection, their response to treatment, and their response to the virus itself.)

Treatment won't get rid of the virus, but it can increase the amount of time between the diagnosis of HIV and diagnosis of AIDS.

Once the diagnosis is changed to AIDS, the body can no longer fight off infections (because the cells the body uses to fight infections have been destroyed by the virus) and any infection, such as a cold, flu, or pneumonia, could be fatal.

These days they have medication that extends their lives for some time I think now, yeah
Obviously their life span is shorter than most people's though :(

It depends if you take your meds prescribed and can afford yeah you could, but if you can't then only expoect 10 or so years.

aweee sad