Editing Linear equation (section)

==Two variables==
A linear equation in two variables {{mvar|x}} and {{mvar|y}} can be written as <math>ax+by+c=0,</math> where {{mvar|a}} and  {{mvar|b}} are not both {{math|0}}.<ref>{{harvnb|Barnett|Ziegler|Byleen|2008|loc = pg. 15}}</ref>

If {{mvar|a}} and {{mvar|b}} are real numbers, it has infinitely many solutions.

===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 therefore has 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. To avoid confusion, the functions whose graph is an arbitrary line are often called ''[[affine function]]s'', and the linear functions such that {{math|1=''c'' = 0}} are often called ''[[linear map]]s''.

===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 or Gradient-intercept form <span class="anchor" id="Slope–intercept form"></span>====
A non-vertical line can be defined by its slope {{mvar|m}}, and its {{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 forms rely on the habit of considering a nonvertical 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 = 0,</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 or Point-gradient form====

A non-vertical line can be defined by its slope {{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>
to emphasize 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 into two different points. The 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 is 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 this 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}}}} (to verify 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_1)=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 by 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>

Besides 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]].