What inherent weaknesses in the political and economic institutions of Rome did the Gracchi brothers' conflicts highlight?
The weakness that this conflict highlighted was the problem of having a large class of poor people who did not have a way to feed themselves and did not have political power.
There were many poor people in the city of Rome at this time in Roman history. Many of them had been pushed into the city by a couple of factors. First, there was the fact that many farmers were evicted from their land so the land could be given to returning soldiers as a reward for their service. Second, there was the fact that land was coming to be held more and more by large landowners who grew lots of food and sold it in the cities. This deprived the rural poor of the chance to get food. For these reasons, they moved to the cities.
Poor and without power, they were unhappy and constituted a potential problem for the Roman system. The conflict between the Gracchi and the Roman elites illustrates this problem.