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If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above. To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that: Problem in words
However, some problems have distinct optimal solutions; for example, the problem of finding a feasible solution to a system of linear inequalities is a linear programming problem in which the objective function is the zero function (i.e., the constant function taking the value zero everywhere).
In the case of two variables, each solution may be interpreted as the Cartesian coordinates of a point of the Euclidean plane. The solutions of a linear equation form a line in the Euclidean plane, and, conversely, every line can be viewed as the set of all solutions of a linear equation in two variables.
In the general case, constraint problems can be much harder, and may not be expressible in some of these simpler systems. "Real life" examples include automated planning, [6] [7] lexical disambiguation, [8] [9] musicology, [10] product configuration [11] and resource allocation. [12] The existence of a solution to a CSP can be viewed as a ...
The dual graph is a representation of how variables are constrained in the dual problem. More precisely, the dual graph contains a node for each dual variable and an edge for every constraint between them. In addition, the edge between two variables is labeled by the original variables that are enforced equal between these two dual variables.
The use of multiple representations supports and requires tasks that involve decision-making and other problem-solving skills. [2] [3] [4] The choice of which representation to use, the task of making representations given other representations, and the understanding of how changes in one representation affect others are examples of such mathematically sophisticated activities.
Computers are usually used to perform the calculations required. With high-speed supercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems. FEM is a general numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems).
Partition problem [2] [3]: SP12 Quadratic assignment problem [3]: ND43 Quadratic programming (NP-hard in some cases, P if convex) Subset sum problem [3]: SP13 Variations on the Traveling salesman problem. The problem for graphs is NP-complete if the edge lengths are assumed integers. The problem for points on the plane is NP-complete with the ...