Graph sample of linear equations.

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.

Linear equations can have one or more variables. Linear equations occur with great regularity in applied mathematics. 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. Linear equations do not include exponents.

Linear equations in two variables[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): y = mx + b,\,

where m and b designate constants. 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 m determines the slope or gradient of that line, and the constant term "b" determines the point at which the line crosses the y-axis, otherwise known as the y-intercept.

Since terms of 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[link]

Linear equations can be rewritten using the laws of elementary algebra into several different forms. These equations are often referred to as the "equations of the straight line." In what follows, x, y, t, and θ are variables; other letters represent constants (fixed numbers).

General form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): Ax + By + C = 0, \,

where A and B are not both equal to zero. The equation is usually written so that A ≥ 0, by convention. The graph 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 (where, 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 (where x is zero), is −C/B, and the slope of the line is −A/B.

Standard form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): Ax + By = C,\,

where A and B are not both equal to zero, A, B, and C are coprime integers, and A is nonnegative (if zero, B must 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. It is worth noting that, while the term occurs frequently in school-level US algebra textbooks, most lines cannot be described by such equations. For instance, the line x + y = √2 cannot be described by a linear equation with integer coefficients since √2 is irrational.

Slope–intercept form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y = mx + b,\,

where m is the slope of the line and b is the y-intercept, which is the y-coordinate of the location where line crosses the y axis. This can be seen by letting x = 0, which immediately gives y = b. It may be helpful to think about this in terms of y = b + mx; where the line originates at (0, b) and extends outward at a slope of m. Vertical lines, having undefined slope, cannot be represented by this form.

Point–slope form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y - y_1 = m( x - x_1 ),\,

where m is the slope of the line and (x₁,y₁) is any point on the line.

The point-slope form expresses the fact that the difference in the y coordinate between two points on a line (that is, Failed to parse (Missing texvc executable; please see math/README to configure.): y - y_1

) is proportional to the difference in the x coordinate (that is, Failed to parse (Missing texvc executable; please see math/README to configure.): x - x_1 ). The proportionality constant is m (the slope of the line).

Two-point form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1),\,

where Failed to parse (Missing texvc executable; please see math/README to configure.): (x_1,y_1)

and Failed to parse (Missing texvc executable; please see math/README to configure.): (x_2,y_2)
are two points on the line with Failed to parse (Missing texvc executable; please see math/README to configure.): x_2
≠ Failed to parse (Missing texvc executable; please see math/README to configure.): x_1

. This is equivalent to the point-slope form above, where the slope is explicitly given as Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{y_2 - y_1}{x_2 - x_1} .

Intercept form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{x}{a} + \frac{y}{b} = 1,\,

where a and b must be nonzero. The graph of the equation has x-intercept a and y-intercept b. The intercept form can be converted to the standard form by setting A = 1/a, B = 1/b and C = 1.

Parametric form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): x = T t + U\,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): y = V t + W.\,

Two simultaneous 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): x = (p - h) t + h\,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): y = (q - k)t + k.\,

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 and other values of t providing extrapolation.

Polar form[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): r=\frac{mr\cos\theta+b}{\sin\theta},

where m is the slope of the line and b is the y-intercept. When θ = 0 the graph will be undefined. The equation can be rewritten to eliminate discontinuities:

Failed to parse (Missing texvc executable; please see math/README to configure.): r\sin\theta=mr\cos\theta+b.\,

Normal form[link]

The normal for a given line is defined to be the shortest segment between the line and the origin. The normal form of the equation of a straight line is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): y \sin \theta + x \cos \theta - p = 0,\,

where θ is the angle of inclination of the normal, and p is the length of the normal. The normal form can be derived from general form by dividing all of the coefficients by

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{|C|}{-C}\sqrt{A^2 + B^2}.

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

2D vector determinant form[link]

The equation of a line can also be written as the determinant of two vectors. If Failed to parse (Missing texvc executable; please see math/README to configure.): P_1

and Failed to parse (Missing texvc executable; please see math/README to configure.): P_2
are unique points on the line, then Failed to parse (Missing texvc executable; please see math/README to configure.): P
will also be a point on the line if the following is true:

Failed to parse (Missing texvc executable; please see math/README to configure.): \det( \overrightarrow{P_1 P} , \overrightarrow{P_1 P_2} ) = 0.

One way to understand this formula is to use the fact that the determinant of two vectors on the plane will give the area of the parallelogram they form. Therefore, if the determinate equals zero then the parallelogram has no area, and that will happen when to vectors are on the same line.

To expand on this we can say that Failed to parse (Missing texvc executable; please see math/README to configure.): P_1 = (x_1 ,\, y_1) , Failed to parse (Missing texvc executable; please see math/README to configure.): P_2 = (x_2 ,\, y_2)

and Failed to parse (Missing texvc executable; please see math/README to configure.): P = (x ,\, y)

. Thus Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{P_1 P} = (x-x_1 ,\, y-y_1)

and Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{P_1 P_2} = (x_2-x_1 ,\, y_2-y_1)

, then the above equation becomes:

Failed to parse (Missing texvc executable; please see math/README to configure.): \det \begin{pmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{pmatrix} = 0.

Thus,

Failed to parse (Missing texvc executable; please see math/README to configure.): ( x - x_1 )( y_2 - y_1 ) - ( y - y_1 )( x_2 - x_1 )=0.

Ergo,

Failed to parse (Missing texvc executable; please see math/README to configure.): ( x - x_1 )( y_2 - y_1 ) = ( y - y_1 )( x_2 - x_1 ).

Then dividing both side by Failed to parse (Missing texvc executable; please see math/README to configure.): ( x_2 - x_1 )

would result in the “Two-point form” shown above, but leaving it here allows the equation to still be valid when Failed to parse (Missing texvc executable; please see math/README to configure.): x_1 = x_2

.

Special cases[link]

Failed to parse (Missing texvc executable; please see math/README to configure.): y = b\,

Horizontal Line y = b

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.

Failed to parse (Missing texvc executable; please see math/README to configure.): x = a\,

Vertical Line x = a

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 a. The slope is undefined. There is no y-intercept, unless a = 0, in which case the graph of the line is the y-axis, and so every real number is a y-intercept.

Failed to parse (Missing texvc executable; please see math/README to configure.): y = y \

and Failed to parse (Missing texvc executable; please see math/README to configure.):  x = x\,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): e = f\,

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). An example would be 3x + 2 = 3x − 5.

Connection with linear functions[link]

A linear equation, written in the form y = f(x) whose graph crosses through the origin, that is whose y-intercept is 0, has the following properties:

Failed to parse (Missing texvc executable; please see math/README to configure.): f ( x_1 + x_2 ) = f ( x_1) + f ( x_2 )\

and

Failed to parse (Missing texvc executable; please see math/README to configure.): f ( a x ) = a f ( x ),\,

where a is any scalar. A function which satisfies these properties is called a linear function (or linear operator, or more generally a linear map). However, linear equations that have non-zero y-intercepts will have neither property above and hence are not linear functions in this sense.

Linear equations in more than two variables[link]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b.

In this form, a₁, a₂, …, a_n are the coefficients, x₁, x₂, …, x_n 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 in n-dimensional Euclidean space (for example, a plane in 3-space).

In vector notation, this can be expressed as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{n} \cdot \overrightarrow{x} = \overrightarrow{n} \cdot \overrightarrow{x_0}

where Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{n}

is a vector normal to the plane, Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{x}
are the coordinates of any point on the plane, and Failed to parse (Missing texvc executable; please see math/README to configure.): \overrightarrow{x_0}
are the coordinates of the origin of the plane.

See also[link]

• Line (geometry)
• Quadratic equation
• Cubic equation
• Quartic equation
• Quintic equation
• Linear inequality
• Linear belief function

References[link]

External links[link]

• Algebraic Equations at EqWorld: The World of Mathematical Equations.
• [1] Video tutorial on solving one step to multistep equations
• Linear Equations and Inequalities Open Elementary Algebra textbook chapter on linear equations and inequalities.

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2007)

In mathematics, a linear map or linear function f(x) is a function which satisfies the following two properties:

• Additivity (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition.
• Homogeneity of degree 1: f(αx) = αf(x) for all α.

It can be shown that additivity implies the homogeneity in all cases where α is rational; this is done by proving the case where α is a natural number by mathematical induction and then extending the result to arbitrary rational numbers. If f is assumed to be continuous as well then this can be extended to show that homogeneity for α any real number, using the fact that rationals form a dense subset of the reals.

In this definition, x is not necessarily a real number, but can in general be a member of any vector space. A less restrictive definition of linear function, not coinciding with the definition of linear map, is used in elementary mathematics.

The concept of linearity can be extended to linear operators. Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian. When a differential equation can be expressed in linear form, it is particularly easy to solve by breaking the equation up into smaller pieces, solving each of those pieces, and adding the solutions up.

Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations (also called linear maps), and systems of linear equations.

The word linear comes from the Latin word linearis, which means pertaining to or resembling a line. For a description of linear and nonlinear equations, see linear equation. Nonlinear equations and functions are of interest to physicists and mathematicians because they can be used to represent many natural phenomena, including chaos.

Integral linearity[link]

For a device that converts a quantity to another quantity there are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of full scale, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.

Many times a device's specifications will simply refer to linearity, with no other explanation as to which type of linearity is intended. In cases where a specification is expressed simply as linearity, it is assumed to imply independent linearity.

Independent linearity is probably the most commonly-used linearity definition and is often found in the specifications for DMMs and ADCs, as well as devices like potentiometers. Independent linearity is defined as the maximum deviation of actual performance relative to a straight line, located such that it minimizes the maximum deviation. In that case there are no constraints placed upon the positioning of the straight line and it may be wherever necessary to minimize the deviations between it and the device's actual performance characteristic.

Zero-based linearity forces the lower range value of the straight line to be equal to the actual lower range value of the device's characteristic, but it does allow the line to be rotated to minimize the maximum deviation. In this case, since the positioning of the straight line is constrained by the requirement that the lower range values of the line and the device's characteristic be coincident, the non-linearity based on this definition will generally be larger than for independent linearity.

For terminal linearity, there is no flexibility allowed in the placement of the straight line in order to minimize the deviations. The straight line must be located such that each of its end-points coincides with the device's actual upper and lower range values. This means that the non-linearity measured by this definition will typically be larger than that measured by the independent, or the zero-based linearity definitions. This definition of linearity is often associated with ADCs, DACs and various sensors.

A fourth linearity definition, absolute linearity, is sometimes also encountered. Absolute linearity is a variation of terminal linearity, in that it allows no flexibility in the placement of the straight line, however in this case the gain and offset errors of the actual device are included in the linearity measurement, making this the most difficult measure of a device's performance. For absolute linearity the end points of the straight line are defined by the ideal upper and lower range values for the device, rather than the actual values. The linearity error in this instance is the maximum deviation of the actual device's performance from ideal.

Linear polynomials[link]

In a different usage to the above, a polynomial of degree 1 is said to be linear, because the graph of a function of that form is a line.

Over the reals, a linear equation is one of the forms:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = m x + b\

where m is often called the slope or gradient; b the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.

Note that this usage of the term linear is not the same as the above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so if and only if b = 0. Hence, if b ≠ 0, the function is often called an affine function (see in greater generality affine transformation).

Boolean functions[link]

In Boolean algebra, a linear function is a function Failed to parse (Missing texvc executable; please see math/README to configure.): f

for which there exist Failed to parse (Missing texvc executable; please see math/README to configure.): a_0, a_1, \ldots, a_n \in \{0,1\}
such that 

Failed to parse (Missing texvc executable; please see math/README to configure.): f(b_1, \ldots, b_n) = a_0 \oplus (a_1 \land b_1) \oplus \cdots \oplus (a_n \land b_n)

for all Failed to parse (Missing texvc executable; please see math/README to configure.): b_1, \ldots, b_n \in \{0,1\}.

A Boolean function is linear if one of the following holds for the function's truth table:

1. In every row in which the truth value of the function is 'T', there are an odd number of 'T's assigned to the arguments and in every row in which the function is 'F' there is an even number of 'T's assigned to arguments. Specifically, f('F', 'F', ..., 'F') = 'F', and these functions correspond to linear maps over the Boolean vector space.
2. In every row in which the value of the function is 'T', there is an even number of 'T's assigned to the arguments of the function; and in every row in which the truth value of the function is 'F', there are an odd number of 'T's assigned to arguments. In this case, f('F', 'F', ..., 'F') = 'T'.

Another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference.

Negation, Logical biconditional, exclusive or, tautology, and contradiction are linear functions.

Physics[link]

In physics, linearity is a property of the differential equations governing many systems; for instance, the Maxwell equations or the diffusion equation.

Linearity of a differential equation means that if two functions f and g are solutions of the equation, then any linear combination af+bg is too.

Electronics[link]

In electronics, the linear operating region of a device, for example a transistor, is where a dependent variable (such as the transistor collector current) is directly proportional to an independent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a high fidelity audio amplifier, which must amplify a signal without changing its waveform. Others are linear filters, linear regulators, and linear amplifiers in general.

In most scientific and technological, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort even a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value, taking it away from the approximately linear part of the transfer function.

Military tactical formations[link]

In military tactical formations, "linear formations" were adapted from phalanx-like formations of pike protected by handgunners towards shallow formations of handgunners protected by progressively fewer pikes. This kind of formation would get thinner until its extreme in the age of Wellington with the 'Thin Red Line'. It would eventually be replaced by skirmish order at the time of the invention of the breech-loading rifle that allowed soldiers to move and fire independently of the large scale formations and fight in small, mobile units.

Art[link]

Linear is one of the five categories proposed by Swiss art historian Heinrich Wölfflin to distinguish "Classic", or Renaissance art, from the Baroque. According to Wölfflin, painters of the fifteenth and early sixteenth centuries (Leonardo da Vinci, Raphael or Albrecht Dürer) are more linear than "painterly" Baroque painters of the seventeenth century (Peter Paul Rubens, Rembrandt, and Velázquez) because they primarily use outline to create shape.^[1] Linearity in art can also be referenced in digital art. For example, hypertext fiction can be an example of nonlinear narrative, but there are also websites designed to go in a specified, organized manner, following a linear path.

Music[link]

In music the linear aspect is succession, either intervals or melody, as opposed to simultaneity or the vertical aspect.

Measurement[link]

In measurement, the term "linear foot" refers to the number of feet in a straight line of material (such as lumber or fabric) generally without regard to the width. It is sometimes incorrectly referred to as "lineal feet"; however, "lineal" is typically reserved for usage when referring to ancestry or heredity.[1] The words "linear"[2] & "lineal" [3] both descend from the same root meaning, the Latin word for line, which is "linea".

See also[link]

• Linear element
• Linear system
• Linear medium
• Linear programming
• Bilinear
• Multilinear
• Linear motor
• Linear A and Linear B scripts.
• Linear interpolation

References[link]

1. ^ Heinrich Wölfflin, Principles of Art History: the Problem of the Development of Style in Later Art, M. D. Hottinger (trans.), Mineola, N.Y.: Dover (1950): pp. 18–72.