Final answer:
The solution to the problems faced by the Republic of Texas included declaring independence, creating a constitution with an American-style judicial system, and not prohibiting slavery, all of which were intended to align Texas with the interests of the United States and paved the way for eventual annexation.
Step-by-step explanation:
A solution to the problems faced by the Republic of Texas involved several key actions. The delegates met in March 1836, declaring independence from Mexico and drafting a constitution that established an American-style judicial system along with an elective presidency and legislature. They also decided that slavery would not be prohibited in Texas, aligning with the interests of wealthy Tejanos and American settlers who supported the notion of Texas eventually becoming part of the United States. This could be seen as solving the immediate political and societal issues they faced within the framework of Mexican law at the time.
The Lone Star Republic then sought to establish itself as an independent nation, securing recognition from countries such as France, Great Britain, Belgium, and the Netherlands. In 1837, the United States recognized Texas as an independent nation. The hope for many was eventual annexation by the United States, which would solve the problem of potential political tensions and insecurity as an independent nation. The eventual annexation of Texas to the Union in 1845 after Sam Houston's presidency not only realized the aspirations of many settlers but also solved lingering issues of Texas's international status and boundary disputes with Mexico, although it also contributed to sparking the Mexican-American War.