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The two circles in the Two points, one line problem where the line through P and Q is not parallel to the given line l, can be constructed with compass and straightedge by: Draw the line m through the given points P and Q. The point G is where the lines l and m intersect; Draw circle C that has PQ as diameter. Draw one of the tangents from G to ...
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).
Watt's curve, which arose in the context of early work on the steam engine, is a sextic in two variables.. One method of solving the cubic equation involves transforming variables to obtain a sextic equation having terms only of degrees 6, 3, and 0, which can be solved as a quadratic equation in the cube of the variable.
An example of a linear function is the function defined by () = (,) that maps the real line to a line in the Euclidean plane R 2 that passes through the origin. An example of a linear polynomial in the variables X , {\displaystyle X,} Y {\displaystyle Y} and Z {\displaystyle Z} is a X + b Y + c Z + d . {\displaystyle aX+bY+cZ+d.}
The kernel may be expressed as the subspace (x, 0) < V: the value of x is the freedom in a solution – while the cokernel may be expressed via the map W → R, (,) (): given a vector (a, b), the value of a is the obstruction to there being a solution. An example illustrating the infinite-dimensional case is afforded by the map f: R ∞ → R ...
Conversely, every line is the set of all solutions of a linear equation. The phrase "linear equation" takes its origin in this correspondence between lines and equations: a linear equation in two variables is an equation whose solutions form a line. If b ≠ 0, the line is the graph of the function of x that
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon; the problem may be eliminated by choosing interpolation points at Chebyshev nodes. [5]
An arrangement of nine points (related to the Pappus configuration) forming ten 3-point lines.. In discrete geometry, the original orchard-planting problem (or the tree-planting problem) asks for the maximum number of 3-point lines attainable by a configuration of a specific number of points in the plane.