Linear equation

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.

Contents

Linear equations in two variables

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

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 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, y1/3, and sin(x) are nonlinear.

Forms for 2D linear equations

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 and t are variables; other letters represent constants (fixed numbers).

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.

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. It is worth noting that, while the term occurs frequently in school-level US algebra textbooks, it makes little mathematical sense since 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

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. Vertical lines, having undefined slope, cannot be represented by this form.

Point–slope form

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 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).

Two-point form

y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1),\,
where (x_1,y_1) and (x_2,y_2) are two points on the line with x_2 ≠ x_1. This is equivalent to the point-slope form above, where the slope is explicitly given as \frac{y_2 - y_1}{x_2 - x_1}.

Intercept form

\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

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.

Polar Form

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. Thus, the equation can be rewritten to eliminate discontinuities:
r\sin\theta=mr\cos\theta+b.\,

Normal form

y \sin \phi + x \cos \phi - p = 0,\,
where \phi 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 is also called the Hesse standard form, after the German mathematician Ludwig Otto Hesse.

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 = 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.
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.

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.

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 linear operator, or more generally a linear map. This property makes linear equations particularly easy to solve and reason about.

Note that not all linear equations are linear functions. A linear function preserves the rules of scalars. In other words, equations that have a zero y-intercept are also linear functions, but equations that have non-zero y-intercepts are not due to the fact that there is no possible way to write a matrix scalar to form the accepted slope form.

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:

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

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 in n-dimensional Euclidean space (for example, a plane in 3-space).

See also

External links

Polynomials
Linear equation · Quadratic function · Cubic function · Quartic function · Quintic equation · Sextic equation · Septic equation