In the 1820s it was shown that quintic equations cannot in general be solved in terms of radicals (see page 1137 ), and by the 1890s it was known that degree 7 equations cannot in general be solved even if elliptic functions are allowed. … The difficulty of solving equations for numerical weather prediction was noted even in the 1920s. … And for example starting in the early 1950s government control of economies based on predictions from linear models became common.

Diophantine equations Any algebraic equation—such as x 3 + x + 1  0 —can readily be solved if one allows the variables to have any numerical value. … Linear Diophantine equations such as a x  b y + c yield simple repetitive results, as in the pictures below, and can be handled essentially just by knowing ExtendedGCD[a, b] .

Linear equations like this were already studied in antiquity. … The picture below shows as a function of a the minimum x that solves the equation.