View source for Linear equation

{{short description|The simplest type of equations}}
{{more citations needed|date=January 2016}}
[[File:Linear Function Graph.svg|thumb|300px|Two graphs of linear equations in two variables]]
In [[mathematics]], a '''linear equation''' is an [[equation]] that may be put in the form
:<math>a_1x_1+\cdots +a_nx_n+b=0,</math>
where <math>x_1, \ldots, x_n</math> are the [[variable (mathematics)|variables]] (or [[unknown (mathematics)|unknowns]] or [[indeterminate (variable)|indeterminates]]), and <math>b, a_1, \ldots, a_n</math> are the [[coefficient]]s, which are often [[real number]]s. The coefficients may be considered as [[parameter]]s of the equation, and may be arbitrary [[expression (mathematics)|expressions]], provided they do not contain any of the variables. To yield a meaningful equation, the coefficient s <math>a_1, \ldots, a_n</math> are required to not be all zero.

In other words, a linear equation is obtained by equating to zero a [[linear polynomial]] over some [[field (mathematics)|field]], from which the coefficients are taken (the symbols used for the variables are supposed to not denote any element of the field).

The [[solution (equation)|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 <math>a_1\ne 0</math>). 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 (geometry)|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 equations. More generally, the solutions of a linear equation in {{mvar|n}} variables form a [[hyperplane]] (a subspace of dimension {{math|''n'' − 1}}) in the [[Euclidean space]] of dimension {{mvar|n}}.

Linear equations occur frequently in all mathematics and their applications in [[physics]] and [[engineering]], partly because [[non-linear system]]s are often well approximated by linear equations.
 
This article considers the case of a single equation with coefficients from the field of [[real number]]s, for which one studies the real solutions. All of its content applies to [[complex number|complex]] solutions and, more generally, for linear equations with coefficients and solutions in any [[field (mathematics)|field]]. For the case of several simultaneous linear equations, see [[system of linear equations]].

== One variable ==
Frequently the term ''linear equation'' refers implicitly to the case of just one variable.

In this case, the equation can be put in the form
:<math>ax+b=0,</math>
and it has a unique solution 
:<math>x=-\frac ba</math>
in the general case where {{mvar|''a'' ≠ 0}}.
In this case, the name ''unknown'' is sensibly given to the variable {{mvar|x}}.

If {{math|1=''a'' = 0}}, there are two cases. Either {{mvar|b}} equals also 0, and every number is a solution. Otherwise {{mvar|''b'' ≠ 0}}, and there is no solution. In this latter case, the equation is said to be [[inconsistent equations|inconsistent]].

==Two variables==
In the case of two variables, any linear equation can be put in the form
:<math>ax+by+c=0,</math>
where the variables are {{mvar|x}} and {{mvar|y}}, and the coefficients are {{mvar|a}}, {{mvar|b}} and {{mvar|c}}.

An equivalent equation (that is an equation with exactly the same solutions) is 
:<math>Ax + By = C,</math>
with {{math|1=''A'' = ''a'', ''B'' = ''b''}}, and {{math|1=''C'' = –''c''}}

These equivalent variants are sometimes given generic names, such as ''general form'' or ''standard form''.<ref>{{harvnb|Barnett|Ziegler|Byleen|2008|loc = pg. 15}}</ref>

There are other forms for a linear equation (see below), which can all be transformed in the standard form with simple algebraic manipulations, such as adding the same quantity to both members of the equation, or multiplying both members by the same nonzero constant.

===Linear function===
{{main|Linear function (calculus)}}
If {{math|''b'' ≠ 0}}, the equation 
:<math>ax+by+c=0</math>
is a linear equation in the single variable {{mvar|y}} for every value of {{mvar|x}}. It has therefore a unique solution for {{mvar|y}}, which is given by 
:<math>y=-\frac ab x-\frac cb.</math>

This defines a [[function (mathematics)|function]]. The [[graph of a function|graph]] of this function is a [[line (geometry)|line]] with [[slope (mathematics)|slope]] <math>-\frac ab</math> and [[y-intercept|{{mvar|y}}-intercept]] <math>-\frac cb.</math> 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 {{math|1=''c'' = 0}}, that is when the line passes through the origin. For avoiding confusion, the functions whose graph is an arbitrary line are often called ''affine functions''.

===Geometric interpretation===
[[File:x is a.svg|thumb|Vertical line of equation {{math|1=''x'' = ''a''}}]]
[[File:y is b.svg|thumb|Horizontal line of equation {{math|1=''y'' = ''b''}}]]
Each solution {{math|(''x'', ''y'')}} of a linear equation
:<math>ax+by+c=0</math> 
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 (geometry)|line]], provided that {{mvar|a}} and {{mvar|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 {{math|''b'' ≠ 0}}, the line is the [[graph of a function|graph of the function]] of {{mvar|x}} that has been defined in the preceding section. If {{math|1=''b'' = 0}}, the line is a ''vertical line'' (that is a line parallel to the {{mvar|y}}-axis) of equation <math>x=-\frac ca,</math> which is not the graph of a function of {{mvar|x}}.

Similarly, if {{math|''a'' ≠ 0}}, the line is the graph of a function of {{mvar|y}}, and, if {{math|1=''a'' = 0}}, one has a horizontal line of equation <math>y=-\frac cb.</math>

===Equation of a line===
There are various ways of defining a line. In the following subsections, a linear equation of the line is given in each case.
 
====Slope–intercept form====
A non-vertical line can be defined by its [[slope (mathematics)]] {{mvar|m}}, and its [[y-intercept|{{mvar|y}}-intercept]] {{math|''y''{{sub|0}}}} (the {{mvar|y}} coordinate of its intersection with the {{mvar|y}}-axis). In this case its ''linear equation'' can be written 
:<math>y=mx+y_0.</math>

If, moreover, the line is not horizontal, it can be defined by its slope and its {{mvar|x}}-intercept {{math|''x''{{sub|0}}}}. In this case, its equation can be written
:<math>y=m(x-x_0),</math>
or, equivalently,
:<math>y=mx-mx_0.</math>

These form rely on the habit of considering a non vertical line as the [[graph of a function]].<ref>{{harvnb|Larson|Hostetler|2007|loc=p. 25}}</ref> For a line given by an equation 
:<math>ax+by+c,</math>
these forms can be easily deduced from the relations 
:<math>\begin{align}
m&=-\frac ab,\\
x_0&=-\frac ca,\\
y_0&=-\frac cb.
\end{align}</math>

====Point–slope form====

A non-vertical line can be defined by its [[slope (mathematics)]] {{mvar|m}}, and the coordinates <math>x_1, y_1</math> of any point of the line. In this case, a linear equation of the line is
:<math>y=y_1 + m(x-x_1),</math>
or 
:<math>y=mx +y_1-mx_1.</math>

This equation can also be written
:<math>y-y_1=m(x-x_1)</math>
for emphasizing that the slope of a line can be computed from the coordinates of any two points.

====Intercept form====
A line that is not parallel to an axis and does not pass through the origin cuts the axes in two different points. The [[y-intercept|intercept values]] {{math|''x''{{sub|0}}}} and {{math|''y''{{sub|0}}}} of these two points are nonzero, and an equation of the line is<ref name=WilsonTracey>{{harvnb|Wilson|Tracey|1925|loc=pp. 52-53}}</ref>
:<math>\frac{x}{x_0} + \frac{y}{y_0} = 1.</math>
(It easy to verify that the line defined by this equation has {{math|''x''{{sub|0}}}} and {{math|''y''{{sub|0}}}} as intercept values).

====Two-point form====
Given two different points {{math|(''x''{{sub|1}}, ''y''{{sub|1}})}} and {{math|(''x''{{sub|2}}, ''y''{{sub|2}})}}, there is exactly one line that passes through them. There are several ways to write a linear equation of is line.

If {{math|''x''{{sub|1}} ≠ ''x''{{sub|2}}}}, the slope of the line is <math>\frac{y_2 - y_1}{x_2 - x_1}.</math> Thus, a point-slope form is<ref name=WilsonTracey />
:<math>y - y_1 = \frac{y_2 - y_1}{x_2 - x_1} (x - x_1).</math>

By [[clearing denominators]], one gets the equation 
:<math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_1)=0,</math>
which is valid also when {{math|1=''x''{{sub|1}} = ''x''{{sub|2}}}} (for verifying this, it suffices to verify that the two given points satisfy the equation).

This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms:
:<math>(y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1) =0</math>
(exchanging the two points changes the sign of the left-hand side of the equation).

====Determinant form====
The two-point form of the equation of a line can be expressed simply in terms of a [[determinant]]. There are two common ways for that.

The equation <math> (x_2 - x_1)(y - y_1) - (y_2 - y_1)(x - x_j)=0</math> is the result of expanding the determinant in the equation
:<math>\begin{vmatrix}x-x_1&y-y_1\\x_2-x_1&y_2-y_1\end{vmatrix}=0.</math>

The equation <math> (y_1-y_2)x + (x_2-x_1)y + (x_1y_2 - x_2y_1)=0</math> can be obtained be expanding with respect to its first row the determinant in the equation
:<math>\begin{vmatrix}
x&y&1\\
x_1&y_1&1\\
x_2&y_2&1
\end{vmatrix}=0.</math>

Beside being very simple and mnemonic, this form has the advantage of being a special case of the more general equation of a [[hyperplane]] passing through {{mvar|n}} points in a space of dimension {{math|''n'' – 1}}. These equations rely on the condition of [[linear dependence]] of points in a [[projective space]].

==More than two variables==
A linear equation with more than two variables may always be assumed to have the form
:<math>a_1 x_1 + a_2 x_2 + \cdots + a_n x_n + b=0.</math>

The coefficient {{mvar|b}}, often denoted {{math|''a''{{sub|0}}}} is called the ''constant term'', sometimes the ''absolute term'',{{citation needed|reason=never heard of this use of "absolute"|date=August 2019}}. Depending on the context, the term ''coefficient'' can be reserved for the {{math|''a''{{sub|''i''}}}} with {{math|''i'' > 0}}.

When dealing with <math>n=3</math> variables, it is common to use <math>x,\; y</math> and <math>z</math> instead of indexed variables.

A solution of such an equation is a {{mvar|n}}-tuples 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. In fact, if every variable has a zero coefficient, then, as mentioned for one variable, the equation is either ''inconsistent'' (for {{mvar|''b'' ≠ 0}}) as having no solution, or all {{nowrap|{{mvar|n}}-tuples}} are solutions.

The {{mvar|n}}-tuples that are solutions of a linear equation in {{nowrap|{{mvar|n}} variables}} are the [[Cartesian coordinates]] of the points of an {{math|(''n'' − 1)}}-dimensional [[hyperplane]] in an {{nowrap|{{mvar|n}}-dimensional}} [[Euclidean space]] (or [[affine space]] if the coefficients are complex numbers or belong to any field). In the case of three variable, this hyperplane is a [[plane (geometry)|plane]].

If a linear equation is given with {{math|''a''{{sub|''j''}} ≠ 0}}, then the equation can be solved for {{math|''x''{{sub|''j''}}}}, yelding 
:<math>x_j = -\frac b{a_j} -\sum_{i\in \{1,\ldots,n\}, i\ne j} \frac {a_i}{a_j}x_i .</math>
If the coefficients are [[real number]]s, this defines a [[real-valued function|real-valued]] [[function of several real variables|function of {{mvar|n}} real variables]].

== See also ==
* [[Linear equation over a ring]]
* [[Algebraic equation]]
* [[Linear inequality]]

== Notes ==
{{Reflist}}

== References ==
* {{citation|first1=R.A.|last1=Barnett|first2=M.R.|last2=Ziegler|first3=K.E.|last3=Byleen|title=College Mathematics for Business, Economics, Life Sciences and the Social Sciences|edition=11th|year=2008|publisher=Pearson|place=Upper Saddle River, N.J.|isbn=0-13-157225-3}}
* {{citation|first1=Ron|last1=Larson|first2=Robert|last2=Hostetler|title=Precalculus:A Concise Course|year=2007|publisher=Houghton Mifflin|isbn=978-0-618-62719-6|url-access=registration|url=https://archive.org/details/precalculusconci00lars}}
* {{citation|first1=W.A.|last1=Wilson|first2=J.I.|last2=Tracey|title=Analytic Geometry|edition=revised|year=1925|publisher=D.C. Heath}}

== External links ==
* {{springer|title=Linear equation|id=p/l059190}}

{{Polynomials}}

[[Category:Elementary algebra]]
[[Category:Equations]]