Linear equation

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Linear equation

A linear equation is an algebraic equation

Algebraic equation

In mathematics, an algebraic equation over a given Field is an equation of the formwhere P and Q are polynomials over that field. For example...
in which each term

Term (mathematics)

The word term is from the Latin terminus which literally means "boundary line, limit", from the Proto-Indo-European root "peg, post, boundary"....
is either a constant or the product of a constant and (the first power of) a single variable

Variable

A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e....
.

Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics

Applied mathematics

Applied mathematics is a branch of mathematics that concerns itself with the mathematical techniques typically used in the application of mathematical knowledge to other domains....
. While they arise quite naturally when modeling many phenomena, they are particularly useful since many non-linear equations may be reduced to linear equations by assuming that quantities of interest vary to only a small extent from some "background" state.

e m and c designate constants (the variable y is multiplied by the constant 1, which as usual is not explicitly written).

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Encyclopedia

A linear equation is an algebraic equation

Algebraic equation

In mathematics, an algebraic equation over a given Field is an equation of the formwhere P and Q are polynomials over that field. For example...
in which each term

Term (mathematics)

Variable

Applied mathematics

Linear equations in two variables

A common form of a linear equation in the two variables and is

where m and c designate constants (the variable y is multiplied by the constant 1, which as usual is not explicitly written). The origin of the name "linear" comes from the fact that the set of solutions of such an equation forms a straight line in the plane. In this particular equation, the constant determines the slope

Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
or gradient of that line; and the constant term

Constant term

In mathematics, the constant term of a polynomial is the term of degree 0. For example, in the polynomialover the variable X, the constant term is 3....
determines the point at which the line crosses the y-axis.

Since terms of a linear equations cannot contain products of distinct or equal variables, nor any power (other than 1) or other function of a variable, equations involving terms such as xy, x², y^1/3, and sin(x) are nonlinear.

Forms for 2D linear equations

Complicated linear equations, such as the ones above, can be rewritten using the laws of elementary algebra

Elementary algebra

Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic....
into several different forms. These equations are often referred to as the "equations of the straight line". In what follows x, y and t are variables; other letters represent constants (fixed numbers).

General form

where A and B are not both equal to zero. The equation is usually written so that A = 0, by convention. The graph

Cartesian coordinate system

In mathematics, the Cartesian coordinate system is used to determine each Point uniquely in a Plane through two numbers, usually called the x-coordinate or abscissa and the y-coordinate or ordinate of the point....
of the equation is a straight line, and every straight line can be represented by an equation in the above form. If A is nonzero, then the x-intercept, that is the x-coordinate of the point where the graph crosses the x-axis (y is zero), is −C/A. If B is nonzero, then the y-intercept, that is the y-coordinate of the point where the graph crosses the y-axis (x is zero), is −C/B, and the slope

Slope

Standard form

Where A, B, and C are the integers whose hugging common factor is 1, A and B are not both equal to zero and, A is non-negative (and if A = 0 then B has to be positive). The standard form can be converted to the general form, but not always to all the other forms if A or B is zero.

Slope–intercept form

Y-axis formula

where m is the slope of the line and c is the y-intercept, which is the y-coordinate of the point where the line crosses the y axis. This can be seen by letting X-axis formula

where m ? 0, is the slope of the line and c is the x-intercept, which is the x-coordinate of the point where the line crosses the x axis. This can be seen by letting , which immediately gives .

Point–slope form

where m is the slope of the line and (x₁,y₁) is any point on the line. The point-slope and slope-intercept forms are easily interchangeable.

The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, ) is proportional to the difference in the x coordinate (that is, ). The proportionality constant is m (the slope of the line).

Intercept form

where h and r must be nonzero. The graph of the equation has r-intercept i and s-intercept h.

The intercept form can be converted to the standard form by setting A = 1/c, B = 1/b and C = 1.

Two-point form

where x₂ ? x₁. The graph passes through the points (x₁,y₁) and (x₂,y₂), and has slope m = (y₂−y₁) / (x₂−x₁).

Parametric form

and

Two simultaneous equations
Simultaneous equations

In mathematics simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations....
in terms of a variable parameter t, with slope m = V / T, x-intercept (VU−WT) / V and y-intercept (WT−VU) / T.

This can also be related to the two-point form, where T = p−h, U = h, V = q−k, and W = k:

and

In this case t varies from 0 at point (h,k) to 1 at point (p,q), with values of t between 0 and 1 providing interpolation
Interpolation

In the mathematics subfield of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....
and other values of t providing extrapolation
Extrapolation

In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty....
.

Normal form

where f is the angle of inclination of the normal and p is the length of the normal. The normal is defined to be the shortest segment between the line in question and the origin. Normal form can be derived from general form by dividing all of the coefficients by

This form is also called the Hesse standard form, after the German mathematician Ludwig Otto Hesse
Otto Hesse

Ludwig Otto Hesse was a Germany mathematician. Hesse was born in K?nigsberg, Kingdom of Prussia, and died in Munich, Kingdom of Bavaria.He worked on algebraic invariants....
.

Special cases

This is a special case of the standard form where A = 0 and B = 1, or of the slope-intercept form where the slope M = 0. The graph is a horizontal line with y-intercept equal to b. There is no x-intercept, unless b = 0, in which case the graph of the line is the x-axis, and so every real number is an x-intercept.

This is a special case of the standard form where A = 1 and B = 0. The graph is a vertical line with x-intercept equal to c. The slope is undefined. There is no y-intercept, unless c = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept.

and

In this case all variables and constants have canceled out, leaving a trivially true statement. The original equation, therefore, would be called an identity

Identity (mathematics)

In mathematics, the term identity has several different important meanings*An identity is an equality that remains true regardless of the values of any variables that appear within it, to distinguish it from an Equality which is true under more particular conditions....
and one would not normally consider its graph (it would be the entire xy-plane). An example is 2x + 4y = 2(x + 2y). The two expressions on either side of the equal sign are always equal, no matter what values are used for x and y.

In situations where algebraic manipulation leads to a statement such as 1 = 0, then the original equation is called inconsistent, meaning it is untrue for any values of x and y (i.e. its graph would be the empty set
Empty set

In mathematics, and more specifically set theory, the empty set is the unique Set having no members. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced....
) An example would be 3x + 2 = 3x − 5.

Connection with linear functions and operators

In all of the named forms above (assuming the graph is not a vertical line), the variable y is a function
Function (mathematics)

The mathematical concept of a function expresses dependence between two quantities, one of which is known and the other which is produced. A function associates a single output to each input element drawn from a fixed Set , such as the real numbers , although different inputs may have the same output....
of x, and the graph of this function is the graph of the equation.

In the particular case that the line crosses through the origin, if the linear equation is written in the form y = f(x) then f has the properties: and

where a is any scalar
Scalar (mathematics)

In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce another vector....
. A function which satisfies these properties is called a linear function, or more generally a linear map. This property makes linear equations particularly easy to solve and reason about.

Linear equations in more than two variables

A linear equation can involve more than two variables. The general linear equation in n variables is:

In this form, a₁, a₂, …, an are the coefficients, x₁, x₂, …, xn are the variables, and b is the constant. When dealing with three or fewer variables, it is common to replace x₁ with just x, x₂ with y, and x₃ with z, as appropriate.

Such an equation will represent an (n–1)-dimensional hyperplane
Hyperplane

A hyperplane is a concept in geometry. It is a higher-dimensional generalization of the concepts of a line in the plane and a plane in 3-dimensional space....
in n-dimensional Euclidean space

Euclidean space

Around 300 Before Christ, the Ancient Greece mathematician Euclid undertook a study of relationships among distances and angles, first in a plane and then in space....
(for example, a plane in 3-space).

External links

at EqWorld: The World of Mathematical Equations.

Linear equation

Linear equations in two variables

Forms for 2D linear equations

General form

Standard form

Slope–intercept form

Y-axis formula

where m ? 0, is the slope of the line and c is the x-intercept, which is the x-coordinate of the point where the line crosses the x axis. This can be seen by letting , which immediately gives .

Point–slope form

Intercept form

Two-point form

Parametric form

Normal form

Special cases

Connection with linear functions and operators

Linear equations in more than two variables

See also

External links