In what fundamental way does the solution set of a system of linear equations differ from the solution set of a system of linear inequalities? Give examples. Discuss the important implications arising from this difference.
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The solution set of a system of linear equations may consist of just one unique solution or infinitely many solutions. In these cases, we say that the system is consistent. However, in some cases the system may not have a solution at all and we say the system is inconsistent.
Consider a system of linear equations that is consistent. The solution(s) consist of ordered pairs of the form (x, y). Every equation of the system holds only for these ordered pairs of x and y, and for no other.
There are several methods of solving a system of linear equations: Addition, Substitution, Cramer's Rule, Matrix method to name a few. The 2-variable system can be solved graphically also, in which case, the solution is given by the point of intersection of the two lines.
The solution set of a system of inequalities consists of a set of points or a solution space, all the points inside which (and in some cases, points lying on the boundary of the region also) ...
Solutions of systems of linear equations and linear inequations have been compared, bringing out the similarities and differences between them. Also presented are an example each for the two types.