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# Linear equation

In mathematics, a linear equation is an equation that may be put in the form where x_1, \ldots, x_n are the variables or unknowns, and c, a_1, \ldots, a_n are coefficients, which are often real numbers, but may be parameters, or even any expression that does not contain the unknowns. 

In algebra an additive map, Z-linear map or additive function is a function that preserves the addition operation: for every pair of elements and in the domain.

## Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

## Affine transformation

In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.

## Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form where P and Q are polynomials with coefficients in some field, often the field of the rational numbers.

## Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

## Coefficient

In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series or any expression; it is usually a number, but may be any expression.

## Complex number

A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.

## Consistent and inconsistent equations

In mathematics and in particular in algebra, a linear or nonlinear system of equations is consistent if there is at least one set of values for the unknowns that satisfies every equation in the system&mdash;that is, that when substituted into each of the equations makes each equation hold true as an identity.

## Constant term

In mathematics, a constant term is a term in an algebraic expression that has a value that is constant or cannot change, because it does not contain any modifiable variables.

## Coordinate system

In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.

## Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

## Elementary algebra

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics.

## Engineering

Engineering is the creative application of science, mathematical methods, and empirical evidence to the innovation, design, construction, operation and maintenance of structures, machines, materials, devices, systems, processes, and organizations.

## Equation

In mathematics, an equation is a statement of an equality containing one or more variables.

## Equation solving

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equality sign.

## Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

## Expression (mathematics)

In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context.

## Extrapolation

In mathematics, extrapolation is the process of estimating, beyond the original observation range, the value of a variable on the basis of its relationship with another variable.

## Factorization of polynomials

In mathematics and computer algebra, factorization of polynomials or polynomial factorization is the process of expressing a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.

## Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

## Gaussian elimination

In linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations.

## Graph of a function

In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.

## Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

## Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

## If and only if

In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

## Interpolation

In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.

## Line (geometry)

The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.

## Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

## Linear belief function

Linear belief functions are an extension of the Dempster–Shafer theory of belief functions to the case when variables of interest are continuous.

## Linear equation over a ring

In algebra, linear equations and systems of linear equations over a field are widely studied.

## Linear function

In mathematics, the term linear function refers to two distinct but related notions.

## Linear function (calculus)

In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph (in Cartesian coordinates with uniform scales) is a line in the plane.

## Linear inequality

In mathematics a linear inequality is an inequality which involves a linear function.

## Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

## Nonlinear system

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.

## Parameter

A parameter (from the Ancient Greek παρά, para: "beside", "subsidiary"; and μέτρον, metron: "measure"), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when identifying the system, or when evaluating its performance, status, condition, etc.

## Physics

Physics (from knowledge of nature, from φύσις phýsis "nature") is the natural science that studies matterAt the start of The Feynman Lectures on Physics, Richard Feynman offers the atomic hypothesis as the single most prolific scientific concept: "If, in some cataclysm, all scientific knowledge were to be destroyed one sentence what statement would contain the most information in the fewest words? I believe it is that all things are made up of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another..." and its motion and behavior through space and time and that studies the related entities of energy and force."Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves."Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of the human intellect in its quest to understand our world and ourselves."Physics is an experimental science. Physicists observe the phenomena of nature and try to find patterns that relate these phenomena.""Physics is the study of your world and the world and universe around you." Physics is one of the oldest academic disciplines and, through its inclusion of astronomy, perhaps the oldest. Over the last two millennia, physics, chemistry, biology, and certain branches of mathematics were a part of natural philosophy, but during the scientific revolution in the 17th century, these natural sciences emerged as unique research endeavors in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms studied by other sciences and suggest new avenues of research in academic disciplines such as mathematics and philosophy. Advances in physics often enable advances in new technologies. For example, advances in the understanding of electromagnetism and nuclear physics led directly to the development of new products that have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.

## Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

## Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

## Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

## Slope

In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.

## System of equations

In mathematics, a set of simultaneous equations, also known as a system of equations or an equation system, is a finite set of equations for which common solutions are sought.

## System of linear equations

In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same set of variables.

## Tax bracket

Tax brackets are the divisions at which tax rates change in a progressive tax system (or an explicitly regressive tax system, although this is much rarer).

## Three-dimensional space

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).

## Two-dimensional space

Two-dimensional space or bi-dimensional space is a geometric setting in which two values (called parameters) are required to determine the position of an element (i.e., point).

## Variable (mathematics)

In elementary mathematics, a variable is a symbol, commonly an alphabetic character, that represents a number, called the value of the variable, which is either arbitrary, not fully specified, or unknown.

## Vertical line test

In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not.

## Y-intercept

In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system.

## References

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