Basic two-variable linear programming problems with numerical solutions and illustrative graphs are available on PurpleMath.com. Wikipedia has more advanced examples represented as pure algebra and a discussion about algorithms that provide general solutions for this class of optimization problem.
In general, a solution to a linear programming problem is the set of variable values for which the problem's linear function is maximized. Problems may have no solution in cases where the problem constraints make this infeasible. A trivially infeasible problem would be one with the constraints "x < 1" and "x > 2," because clearly no value of x could satisfy this.
Algorithms that provide general solutions to linear programming problems, such as the simplex method, are detailed on Wikipedia. These are often used to solve problems involving numerous constraints and variables, such as in solvers for constraint-based user interface layouts in computer software.