According to the Buddha, how can we eliminate suffering?

By seeing clearly that our suffering is rooted in an attachment to a construct of self that doesn't exist (Annata). Buddhism is not about supressing desire or getting rid of it, as some wrongly think, but about understanding what causes attachment and doing something about the root cause. As the Buddhist teacher Ajahn Sumedho puts it,

"The actual word Buddha means Awakened. It’s a “wake-up” teaching. Somebody asked me one time, “Could you describe Buddhism in one sentence?” And I said, “I can do it in one word.” He said, “What’s that?” “Wake-up!” This isn’t Page 4 of 17

about me waking up, but it’s an invitation to pay attention, to be open, receptive, here and now, in which the sense of your thinking process, ego, cultural assumptions, the thought process itself, recedes. It’s not about getting rid of desire, or getting rid of your self; but of not limiting yourself to the language, the thoughts, the memories, the identities that we tend to when we’re not fully awake. Taking refuge in Buddha, Dhamma, Sangha, on a practical level, a functional level, is this waking up."

- Awakened Consciousness - A Dhamma talk given by Ajahn Sumedho at Abhayagiri Buddhist Monastery on the Lunar Observance Night, June 18, 2008

The cause of suffering is desire.

So you must eliminate desire from your life.

By eliminating desire, exacly as haoiez said. This comes with discipline, practice, and I imagine misery for a while

Suffering is eliminated according to Buddhism doctrine by relinquishing attachment to matters, conditions and people considered changeable, temporal and transient. Most viable religious and philosophical doctrines and principles cite attachment as the cause of emotional and spiritual suffering. This belief is based upon the principle and line of thinking that mankind must free and liberate the mind from the incumbent baggage both emotional and material to be truly free in will and mind. Fear whether of hunger, loneliness, loss, entrapment, poverty causes undue attachment and a depletion of spiritual strength by the doing so. Hatred of a person, a people, a situation or belief causes the erection of barriers between reason and actuality, inciting irrational behaviour and closed mindedness liable to lead to negative thinking. Ignorance causes uncertainty, insecurity, lack of faith, insular blinkered thinking in defence of perceived aggressors, debilitating peace, love and harmony. When hate itself is cured with understanding of its cause and triggers one is able to rise above nihilistic, mentally constricting, spiritually oppressive modes of thinking. On the basis that if one is unable or limited in ability to change ones world and the thinking and behaviour of those about us, then to transcend intellectually and spiritually frees us from the outcome, consequence of negative actions and travails of others.

The teachings of the Buddhist doctrine produces supremely intelligent and well balanced thinking individuals, and the only religion or belief system I would recommend to any truth seeker, in this age of widely available and readily accessible data on all faiths and belief system son internets and libraries abundant with gurus, great thinkers and concise explanations of the fundamentals of truth and wisdom.

Never get married!

1) "According to the Pali Tipitaka, the Four Noble Truths were the first teaching of Gautama Buddha after attaining Nirvana. They are sometimes considered as containing the essence of the Buddha's teachings and are presented in the manner of a medical diagnosis and remedial prescription – a style common at that time:
1. Life as we know it ultimately is or leads to suffering (dukkha) in one way or another.
2. Suffering is caused by craving or attachments to worldly pleasures of all kinds. This is often expressed as a deluded clinging to a certain sense of existence, to selfhood, or to the things or phenomena that we consider the cause of happiness or unhappiness.
3. Suffering ends when craving ends, when one is freed from desire. This is achieved by eliminating all delusion, thereby reaching a liberated state of Enlightenment (bodhi);
4. Reaching this liberated state is achieved by following the path laid out by the Buddha."
Source and further information:
http://en.wikipedia.org/wiki/Buddhism#Suffering:_Causes_and_Solution


2) ""The Noble Truth of Suffering (dukkha), monks, is this: Birth is suffering, aging is suffering, sickness is suffering, death is suffering, association with the unpleasant is suffering, dissociation from the pleasant is suffering, not to receive what one desires is suffering — in brief the five aggregates subject to grasping are suffering.

"The Noble Truth of the Origin (cause) of Suffering is this: It is this craving (thirst) which produces re-becoming (rebirth) accompanied by passionate greed, and finding fresh delight now here, and now there, namely craving for sense pleasure, craving for existence and craving for non-existence (self-annihilation).

"The Noble Truth of the Cessation of Suffering is this: It is the complete cessation of that very craving, giving it up, relinquishing it, liberating oneself from it, and detaching oneself from it.

"The Noble Truth of the Path Leading to the Cessation of Suffering is this: It is the Noble Eightfold Path, and nothing else, namely: right understanding, right thought, right speech, right action, right livelihood, right effort, right mindfulness and right concentration."
Source and further information:
http://www.accesstoinsight.org/tipitaka/sn/sn56/sn56.011.piya.html

through the eight fold path