Linear equation

In mathematics, a linear equation is an equation that may be put in the form \(a_1x_1+\ldots+a_nx_n+b=0,\) where \(x_1,\ldots,x_n\) are the variables (or unknowns), and \(b,a_1,\ldots,a_n\) 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 \(a_1, \ldots, a_n\) 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 \(a_1\ne 0\)). 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 equation. 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, to linear equations with coefficients and solutions in any field. For the case of several simultaneous linear equations, see system of linear equations.

A linear equation in two variables x and y can be written as \(ax+by+c=0,\) where a and b are not both 0.

If a and b are real numbers, it has infinitely many solutions.

If b ≠ 0, the equation

\(ax+by+c=0\)

is a linear equation in the single variable y for every value of x. It therefore has a unique solution for y, which is given by

\(y=-\frac ab x-\frac cb.\)

This defines a function. The graph of this function is a line with slope \(-\frac ab\) and y-intercept \(-\frac cb.\) 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. To avoid confusion, the functions whose graph is an arbitrary line are often called affine functions, and the linear functions such that c = 0 are often called linear maps.

Each solution (x, y) of a linear equation

\(ax+by+c=0\)

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 \(x=-\frac ca,\) 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 \(y=-\frac cb.\)

A linear equation with more than two variables may always be assumed to have the form

\(a_1 x_1 + a_2 x_2 + \cdots + a_n x_n + b=0.\)

The coefficient b, often denoted a₀ is called the constant term (sometimes the absolute term in old books). Depending on the context, the term coefficient can be reserved for the a_i with i > 0.

When dealing with \(n=3\) variables, it is common to use \(x,\; y\) and \(z\) instead of indexed variables.

A solution of such an equation is a n-tuple 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. 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 variables, this hyperplane is a plane.

If a linear equation is given with a_j ≠ 0, then the equation can be solved for x_j, yielding

\(x_j = -\frac b{a_j} -\sum_{i\in \{1,\ldots,n\}, i\ne j} \frac {a_i}{a_j}x_i .\)

Condensed: the full section is in Wikipedia.