The issue arises from the input format in Mathematica 5 when attempting to solve the equations "ax + by == 0" and "cx + dy == 1". The output "{{}}" indicates that the system did not find a solution, likely due to the incorrect interpretation of the variables. It is crucial to ensure proper spacing between coefficients and variables, such as using "a x" instead of "ax". This distinction allows Mathematica to correctly interpret the expression as a product of "a" and "x", which is essential for obtaining the desired solution.