Linear equation

In mathematics, a linear equation is an equation that may be put in the form

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 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 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

An equivalent equation (that is an equation with exactly the same solutions) is

These equivalent variants are sometimes given generic names, such as general form or standard form.[1]

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.

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

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.

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.[2] For a line given by an equation

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[3]

(It is 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.

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.

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.

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.