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In mathematics, a linear equation is an equation that may be put in the form
where are the variables (or unknowns), and are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be arbitrary expressions, provided they do not contain any of the variables. To yield a meaningful equation, the coefficients are required to not all be zero.
Alternatively a linear equation can be obtained by equating to zero a linear polynomial over some field, from which the coefficients are taken.
The solutions of such an equation are the values that, when substituted for the unknowns, make the equality true.
In the case of just one variable, there is exactly one solution (provided that ). Often, the term linear equation refers implicitly to this particular case, in which the variable is sensibly called the unknown.
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. This is the origin of the term linear for describing this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (a subspace of dimension n − 1) in the Euclidean space of dimension n.
Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.
This article considers the case of a single equation with coefficients from the field of real numbers, for which one studies the real solutions. All of its content applies to complex solutions and, more generally, for linear equations with coefficients and solutions in any field. For the case of several simultaneous linear equations, see system of linear equations.
Frequently the term linear equation refers implicitly to the case of just one variable.
In this case, the equation can be put in the form
and it has a unique solution
in the general case where a ≠ 0. In this case, the name unknown is sensibly given to the variable x.
If a = 0, there are two cases. Either b equals also 0, and every number is a solution. Otherwise b ≠ 0, and there is no solution. In this latter case, the equation is said to be inconsistent.
In the case of two variables, any linear equation can be put in the form
where the variables are x and y, and the coefficients are a, b and c.
An equivalent equation (that is an equation with exactly the same solutions) is
with A = a, B = b, and C = –c
These equivalent variants are sometimes given generic names, such as general form or standard form.
There are other forms for a linear equation (see below), which can all be transformed in the standard form with simple algebraic manipulations, such as adding the same quantity to both members of the equation, or multiplying both members by the same nonzero constant.
If b ≠ 0, the equation
is a linear equation in the single variable y for every value of x. It has therefore a unique solution for y, which is given by
This defines a function. The graph of this function is a line with slope and y-intercept The functions whose graph is a line are generally called linear functions in the context of calculus. However, in linear algebra, a linear function is a function that maps a sum to the sum of the images of the summands. So, for this definition, the above function is linear only when c = 0, that is when the line passes through the origin. For avoiding confusion, the functions whose graph is an arbitrary line are often called affine functions.
Each solution (x, y) of a linear equation
may be viewed as the Cartesian coordinates of a point in the Euclidean plane. With this interpretation, all solutions of the equation form a line, provided that a and b are not both zero. 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 has been defined in the preceding section. If b = 0, the line is a vertical line (that is a line parallel to the y-axis) of equation which is not the graph of a function of x.
Similarly, if a ≠ 0, the line is the graph of a function of y, and, if a = 0, one has a horizontal line of equation
There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
A non-vertical line can be defined by its slope m, and its y-intercept y0 (the y coordinate of its intersection with the y-axis). In this case its linear equation can be written
If, moreover, the line is not horizontal, it can be defined by its slope and its x-intercept x0. In this case, its equation can be written
These forms rely on the habit of considering a non vertical line as the graph of a function.For a line given by an equation
these forms can be easily deduced from the relations
A non-vertical line can be defined by its slope m, and the coordinates of any point of the line. In this case, a linear equation of the line is
This equation can also be written
for emphasizing that the slope of a line can be computed from the coordinates of any two points.
A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The intercept values x0 and y0 of these two points are nonzero, and an equation of the line is
(It easy to verify that the line defined by this equation has x0 and y0 as intercept values).
Given two different points (x1, y1) and (x2, y2), there is exactly one line that passes through them. There are several ways to write a linear equation of this line.
If x1 ≠ x2, the slope of the line is Thus, a point-slope form is
By clearing denominators, one gets the equation
which is valid also when x1 = x2 (for verifying this, it suffices to verify that the two given points satisfy the equation).
This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:
(exchanging the two points changes the sign of the left-hand side of the equation).
The two-point form of the equation of a line can be expressed simply in terms of a determinant. There are two common ways for that.
The equation is the result of expanding the determinant in the equation
The equation can be obtained be expanding with respect to its first row the determinant in the equation
Beside being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a hyperplane passing through n points in a space of dimension n – 1. These equations rely on the condition of linear dependence of points in a projective space.
A linear equation with more than two variables may always be assumed to have the form
The coefficient b, often denoted a0 is called the constant term, sometimes the absolute term,[ citation needed ]. Depending on the context, the term coefficient can be reserved for the ai with i > 0.
When dealing with variables, it is common to use and instead of indexed variables.
A solution of such an equation is a n-tuples such that substituting each element of the tuple for the corresponding variable transforms the equation into a true equality.
For an equation to be meaningful, the coefficient of at least one variable must be non-zero. In fact, if every variable has a zero coefficient, then, as mentioned for one variable, the equation is either inconsistent (for b ≠ 0) as having no solution, or all n-tuples are solutions.
The n-tuples that are solutions of a linear equation in n variables are the Cartesian coordinates of the points of an (n − 1)-dimensional hyperplane in an n-dimensional Euclidean space (or affine space if the coefficients are complex numbers or belong to any field). In the case of three variable, this hyperplane is a plane.
If a linear equation is given with aj ≠ 0, then the equation can be solved for xj, yielding
If the coefficients are real numbers, this defines a real-valued function of n real variables.
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.
In mathematics, an equation is a statement that asserts the equality of two expressions, which are connected by the equals sign "=". The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any equality is an equation.
In mathematics, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.
In algebra, a quadratic equation is any equation that can be rearranged in standard form as
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve.
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point, a line and three-dimensional space. Planes can arise as subspaces of some higher-dimensional space, as with one of a room's walls, infinitely extended, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry.
In mathematics, a system of linear equations is a collection of one or more linear equations involving the same set of variables. For example,
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms of the same function are given; each further term of the sequence or array is defined as a function of the preceding terms of the same function.
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748.
In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
In mathematics, a cubic function is a function of the form
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
In mathematics, an algebraic equation or polynomial equation is an equation of the form
In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Monomials – relationships of the form – appear as straight lines in a log–log graph, with the power term corresponding to the slope, and the constant term corresponding to the intercept of the line. Thus these graphs are very useful for recognizing these relationships and estimating parameters. Any base can be used for the logarithm, though most commonly base 10 are used.
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say, or etc.. It is a two-dimensional case of the general n-dimensional phase space.
In geometry, the trilinear coordinatesx:y:z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio x:y is the ratio of the perpendicular distances from the point to the sides opposite vertices A and B respectively; the ratio y:z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z:x and vertices C and A.
In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:. It shows the data which is not equal in graph form.
In mathematics, the characteristic equation is an algebraic equation of degree n upon which depends the solution of a given nth-order differential equation or difference equation. The characteristic equation can only be formed when the differential or difference equation is linear and homogeneous, and has constant coefficients. Such a differential equation, with y as the dependent variable, superscript (n) denoting nth-derivative, and an, an − 1, ..., a1, a0 as constants,
In calculus and related areas of mathematics, a linear function from the real numbers to the real numbers is a function whose graph is a line in the plane. The characteristic property of linear functions is that when the input variable is changed, the change in the output is proportional to the change in the input.