NationMaster - Encyclopedia: Linear equation

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, two, three or more variables. A common form of a linear equation in the two variables $x$ and $y$ is Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In elementary mathematics, a term is either a single number or variable, or the product of several numbers and/or variables. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ... In computer science and mathematics, a variable (pronounced ) (sometimes called an object or identifier in computer science) is a symbolic representation used to denote a quantity or expression. ...

$y = mx + b,,$

where m and b designate constants (the variable y is multiplied by the constant 1, which as usual is not explicitly written). The set of solutions of such an equation forms a straight line in the plane, which is the origin of the name "linear". 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. This article is about the mathematical term. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

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. In mathematics, a nonlinear system is a system which is not linear i. ...

Graph sample of linear equations.

1 Forms for 2D linear equations
2 Connection with linear functions and operators
3 Linear equations in more than two variables
4 See also
5 References
6 External links

Image File history File links FuncionLineal02. ... Image File history File links FuncionLineal02. ...

Forms for 2D linear equations

Complicated linear equations, such as the ones above, can be rewritten using the laws of elementary algebra into several simpler forms. In what follows x, y and t are variables; other letters represent constants (unspecified but fixed numbers). 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. ... In mathematics and the mathematical sciences, a constant is a fixed, but possibly unspecified, value. ...

General form

$Ax + By + C = 0,,$

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 (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 of the line is −A/B.

Line redirects here. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... This article is about the mathematical term. ...

Standard form

$Ax + By = C,,$

where A, B, and C are integers whose greatest 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

$y = mx + b,,$

where m is the slope of the line and b 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 = 0

, which immediately gives

y = b

X-axis formula

$x = frac{y}{m} + c,,$

where m 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

y = 0

, which immediately gives

x = c

Point–slope form

$y - y_1 = m cdot ( x - x_1 ),$

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,

y - y 1

) is proportional to the difference in the x coordinate (that is,

x - x 1

). The proportionality constant is m (the slope of the line).

Intercept form

$frac{x}{c} + frac{y}{b} = 1$

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

Two-point form

$y - k = frac{q - k}{p - h} (x - h),$

where p ≠ h. The graph passes through the points (h,k) and (p,q), and has slope m = (q−k) / (p−h).

Parametric form

$x = T t + U,$

and

$y = V t + W.,$

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:

$x = (p - h) t + h,$

and

$y = (q - k)t + k.,$

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.

In Mathematics simultaneous equations are a set of equations containing multiple variables. ... For other uses, see Interpolation (disambiguation). ... In mathematics, extrapolation is the process of constructing new data points outside a discrete set of known data points. ...

Normal form

$y sin phi + x cos phi - p = 0,,$

where φ 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 $frac{|C|}{-C}sqrt{A^2 + B^2}$ . This form also called Hesse standard form, named after a German mathematician Ludwig Otto Hesse.

Ludwig Otto Hesse (22 April 1811 â€“ 4 August 1874) was a German mathematician. ...

Special cases

$y = b.,$

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.

$x = c.,$

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.

$y = y$ and $x = x.,$

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.

$e = f.,$

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.

In mathematics, the term identity has several important uses: 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. ... The empty set is the set containing no elements. ...

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 of x, and the graph of this function is the graph of the equation. This article is about functions in mathematics. ...

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:

$f ( x + y ) = f ( x ) + f ( y ),$

and

$f ( a x ) = a f ( x ),,$

where a is any scalar. 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. 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. ... In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

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. 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. ... To do: 20th century mathematics chaos theory, fractals Lyapunov stability and non-linear control systems non-linear video editing See also: Aleksandr Mikhailovich Lyapunov Dynamical system External links http://www. ...

Linear equations in more than two variables

Main article: System of linear equations

A linear equation can involve more than two variables. The general linear equation in n variables is: In mathematics and linear algebra, a system of linear equations is a set of linear equations such as A standard problem is to decide if any assignment of values for the unknowns can satisfy all three equations simultaneously, and to find such an assignment if it exists. ...

$a_1 x_1 + a_2 x_2 + cdots + a_n x_n = b.$

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). A hyperplane is a concept in geometry. ... Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called Euclidean geometry, which is the study of the relationships between angles and distances in space. ...

References

This article does not cite any references or sources. (May 2008)
Please help improve this article by adding citations to reliable sources. Unverifiable material may be challenged and removed.

External links

Algebraic Equations at EqWorld: The World of Mathematical Equations.

Categories: Elementary algebra | Equations

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