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Two Graphs of linear equations in two variables

In mathematics, a linear equation is an equation that may be put in the form

where are the variables (or unknowns or indeterminates), and are the coefficients, which are often real numbers. The coefficients may be considered as parameters of the equation, and may be stated as arbitrary expressions, restricted to not contain any of the variables. To yield a meaningful equation for non-zero values of the coefficients are required not to be all zeros.

In the words of algebra, a linear equation is obtained by equating to zero a linear polynomial over some field, where the coefficients are taken from, and that does not contain the symbols for the indeterminates.

The solutions of such an equation are the values that, when substituted to the unknowns, make the equality true.

The case of just one variable is of particular importance, and it is frequent that the term linear equation refers implicitly to this particular case, in which the name unknown for the variable is sensibly used.

All the pairs of numbers that are solutions of a linear equation in two variables form a line in the Euclidean plane, and every line may be defined as the solutions of a linear equation. This is the origin of the term linear for qualifying this type of equations. More generally, the solutions of a linear equation in n variables form a hyperplane (of dimension n – 1) in the Euclidean space of dimension n.

Linear equations occur frequently in all mathematics and their applications in physics and engineering, partly because non-linear systems are often well approximated by linear equations.

This article considers the case of a single equation with coefficients from the field of real numbers, for which one studies the real solutions. All its content applies for complex solutions and, more generally, for linear equations with coefficient and solutions in any field. For the case of several simultaneous linear equations, see System of linear equations.

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One variableEdit

Frequently the term linear equation refers implicitly to the case of just one variable. This case, in which the name unknown for the variable is sensibly used, is of particular importance, since it offers a unique value as solution to the equation. According to the above definition such an equation has the form

 

and, for a ≠ 0, a unique value as solution

 

The above equation may always be rewritten to

 

and the solution is of course the same in both cases:

 

In the case of  , two possibilities emerge:

  1.   Every value for   is a solution to the equation   and
  2.   There is no solution for the equation   the equation is said to be inconsistent.

Two variablesEdit

In the case of just two variables the indexed variable names   and   and the respective coefficients   and   are often replaced, for the convenience of not having to deal with indices, by  ,  ,   and  , respectively. As a consequence, the so called constant term, named coefficient   in the above notation, must also be renamed;   suggests itself. A linear equation in two variables is then denoted as

 

Any change to such an equation that does not alter the set of solutions, i.e., the set of pairs  , that satisfy this equation (i.e., make it an identity), generates an equivalent equation. It is immediate that changing the involved names (e.g. capitalizing names or using other letters) and also reordering the equation (e.g. by moving terms to the other side), does not change this set of solutions, and thus results in an equivalent equation, like, e.g.

  with   and  

These equivalent variants are sometimes given generic names, like general form or standard form,[1] but contribute no new concepts.

The set of solutions also does not change when both sides of the equation are multiplied by the same non-zero number. According to the above definition,   and   (identically   and  ) are not both zero, so multiplying the equation by the reciprocal of one of these non-zero coefficients, results in an equivalent equation with   as the coefficient of one variable. This variable can be isolated on the left hand side, leaving an expression, possibly containing the other variable on the right hand side. This leads to either

  with   and   or
  with   and  

When both coefficients   and   are not zero, then both forms exist, and, assuming real numbers as coefficients and for the domain of the variables, the set of solutions for both equations can then be denoted as

  which is equal to the set  

In this case both components of the pairs in the set   vary over all real numbers, thereby depending in a so called linear affine manner on the respective other.

When exactly one coefficient, either   or  , is not zero, then one equation remains, which is either

  for   with the set of solutions   or
  for   with the set of solutions  

For both alternatives this is a set of pairs of numbers, where either the second component is a constant, and the first varies over all the reals ( ), or the first is a constant, and the second varies over all the reals ( ).

In Cartesian coordinatesEdit

Every single solution of a linear equation in two variables can be interpreted as two coordinate values, fixing a point in the Euclidean plane with a Cartesian coordinate system. The sets of solutions of such an equation make up a two-dimensional graph, which can be depicted in this plane. Conventionally, the first component of a solution  , the  -value, is assigned to a horizontally drawn  -axis, and the second component, the  -value, to a vertical  -axis.

 
Vertical Line  
 

In the case of   the equation is   and the set of its solutions   has a vertical line as its graph, as shown in the figure to the right. The value   where the line intersects the  -axis in the point  , is called an  -intercept. Except for   when the graph coincides with the  -axis, graphs of this kind do not intersect the  -axis, they have no  -intercept.

The set of solutions defines a function   and, simultaneously, the graph of this function, by interpreting the pairs   as   provided that any two such solutions that differ in their second value ( ), also differ in their respective first values ( ). The set   violates this condition: all real values   in the second component have the same first component   Nevertheless, a graph for this set may be drawn, but it is not a graph of a function under the conventional assignment of axes, it obviously fails the vertical line test. This is the only type of straight line which is not the graph of any function  .

 
Horizontal Line  
 

The sets   and   satisfy the above condition, and the graph of   is shown to the right. In this case of   the graph of the constant function   is a horizontal line. The value   where the line intersects the  -axis, is called  -intercept. Except for   where the graph coincides with the  -axis, graphs of this kind have no  -intercept.

In the case of   with the equation   the set of solutions is   It consists of pairs of numbers, with the first component varying over all the reals, and the other being calculated by a simple expression, representing a linear map ( ) and adding a constant ( ). This is sometimes called a linear affine function, or simply also linear function, slightly abusing the strict term linear. Also in this case the graph of a linear equation in two variables is a straight line (see figure at the top) that intersects the  -axis at  -intercept   (i.e.,   is a solution) and the  -axis at the  -intercept   (i.e.,   is a solution).

Besides the intercepts being obvious from graphing the solutions of a linear equation in two variables, also their ratio (if it exists) can be graphically interpreted as determining the incline of the considered line (and all lines parallel to it). The slope of a straight line, usually introduced as rise over run, is the negative ratio of the rise, the  -intercept, to the run, the  -intercept. The negative sign accommodates for a positive slope, when the line rises for increasing  -values. Immediately

 

which holds if both intercepts exist. If the  -intercept does not exist ( ), the slope   equals   belonging to a horizontal line.

Since rise and run of a straight line can be determined not only between the intercept points and the origin (  and  ), but also between arbitrary points   and   on the line, the slope may also be determined by

 

Denoting the angle enclosed by the  -axis and the line as   then

 

For   the slope is undefined ( ).

This shows that only two of   and   can be selected independently.

With the premise that at least one axis is intersected, and since both intercept values may range over the whole real number line, all parallels to both axes as well as all oblique straight lines, i.e., in fact all straight lines in the Euclidean plane, can be expressed by linear equations in two variables, and all such equations denote either oblique or axis-parallel straight lines. Therefore all equations, equivalent to one of the above forms are often referred to as "equations of a line". They are adjusted to fit best to specific tasks, and are given therefore specific names, described below. In what follows,   are the names of variables, and other letters denote constants (fixed numbers) as coefficients.

Slope–intercept formEdit

This form relies on the habit of writing   and the conventional way of assigning the variables of the linear equation to the axes of a Cartesian coordinate system, drawn in the conventional manner as described above. This form exists only for   allowing to isolate   on the left hand side

 

This way the slope   describes the inclination of the straight line which is the graph of this equation. The slope is positive for a line ascending to the right and negative, if the line ascends to the left. A zero-slope   belongs to a horizontal line.

The  -intercept   fixes the point   where the line crosses the  -axis, and   characterizes lines that cross the origin  

Recalling the  -intercept as   the above slope-intercept form, employing the slope   and the  -intercept, can be transformed to

 

involving the slope   and the  -intercept  .

In the case of   there is no slope-intercept form in the above way, because a slope does not exist for  .

For   it is possible to express the inverse functions   in the slope-intercept form as

  with  

The graph of this equation, having the same set of solutions, is necessarily identical to the above graph, but depicting it under exchanged assignment of the variables to the coordinate-axes ( ), yields the usual  -graph for inverse functions, the  -graph mirrored along   This holds for both   and  

The graph of a vertical line ( ) with no existing slope and the equation   changes under this inverted assignment to the graph of the function   with zero-slope (  an arbitrary constant), and vice versa.

Point–slope formEdit

It is observational evident that fixing an arbitrary point on a line and a slope uniquely defines this straight line. In the slope-intersect form this point on the line is either taken as the intersection   with the  -axis, or the intersection   with the  -axis and is combined with the slope  , provided its existence, to establish the equation for the according line. Generalizing this approach to an arbitrary point with coordinates   yields:

 

The point-slope form expresses the fact that the difference of the   coordinates between two points on a line (i.e.,  ) is proportional to the difference of the  -coordinate (i.e.,  ), with the proportionality constant   the slope of the line.

Intercept formEdit

For a straight line that crosses both coordinate axes outside the origin, both intercept values exist and are non-zero. This implies that also   is nonzero, and such lines can be specified via the intercept form, that employs these two intercept values to specify an appropriate equation

 

The intercept form results from moving   in the equation   to the right side, and then multiplying both sides of the equation with   yielding

 

which is identical to the above form. The intercept form also works conveniently in higher dimensions for specifying (hyper)planes, when their intersections with all coordinate axes exist and are known.

Two-point formEdit

Two points   and   with   (no vertical lines!) determine the slope of the line through these points. This slope, calculated as above, can be used with either point to employ the point-slope form, thereby establishing appropriate equations for this line, based on two points with different  -values. This yields

  for  

In the rest of this paragraph   is used.

Expanded formEdit

Expanding, regrouping, and appropriately factoring the products leads to

 

identifying:   and  

Symmetric formEdit

Multiplying both sides of the 2-point form by   yields an equation in a symmetric form

 

This form also works in the case of a non-existing slope ( ), but requires   in this case: it correctly delivers  

Determinant formEdit

The products in the above equation result also from the evaluation of a 2-rowed determinant, inducing this form of the linear equation:

 

Mnemonic determinantEdit

The products on the left hand side of the expanded version can be reproduced by evaluating the 3-rowed determinant, designed for easy memorability:

 

Vectorial treatmentEdit

Any pair of numbers   may be treated as a vector, given by two components with respect to a Cartesian coordinate system. A (naive) vector starts at the origin  , and ends at the given coordinates. Any two non-collinear vectors   and   span a parallelogram, with these three points. The area   of this parallelogramm can be calculated as the magnitude of the exterior product (2dim-cross product, geometric product, ...) of these vectors. In components this can be done by evaluating the absolute value of a determinant with the components:

 

Two given points   and an arbitrary third point   are on one straight line (collinear), if, e.g., the vector from   to   and the vector from   to   span no parallelogram, i.e., a parallelogram with area zero, i.e., also the vectors are collinear.

The vector from point   to point   can be expressed as

 

and similarly the vector from point   to an arbitrary point   is

 

Equating the exterior product of these two vectors, as specified above, to zero, yields a linear equation

 

which is identical to the determinant form above.

Matrix formEdit

Writing a linear equation in two unknowns in the form

 

and considering the collection of coefficients   as a  -matrix, and the collection of variables   as a  -matrix, then their matrix product equals the  -matrix  

 

This notation can easily expanded to more linear equations in more than two variables. For example, a system of two equations in two variables

 
 

can be denoted with a  -matrix and a  -matrix for the coefficients, by equaling the matrix product of the  -coefficient matrix with the  -variable matrix to the  -matrix of the constant terms:

 

A system of three linear equations in four variables would employ a  -matrix for the coefficients of the variables, which, multiplied with the  -(column)-matrix of the variables, is equaled to the  -matrix of the constant terms. For this ready extendability to higher dimensions, the matrix notation is a common representation tool for a system of linear equations, in linear algebra, and in computer programming. There are named methods for solving systems of linear equations, like Gauss-Jordan which can be expressed in matrix elementary row operations.

Parametric formEdit

The parametric form of a curve is useful to e.g. describe the movement of a point along this curve, and controlling this movement with a single parameter. This setting resembles the task in physics, where a particle starts at time   at some point in space, say  , and travels along the curve, where it reaches point   at time   With linear equations the curves are restricted to straight lines.

This task can be solved by adding a motion from   in the direction of the  -axis and a simultaneous motion from   in the direction of the  -axis, both motions controlled by the parameter   The motion in  -direction can be described as

 

and similarly, the motion in  -direction can be described as

 

These two linear equations, with variables   and  , make up a parametric form for a linear equation with variables   that can be constructed from the two-point form with   and   as points.

For   and for   For all   in the interval   the point   is on the straight line segment connecting the points for   and   This is sometimes called interpolation. For values of   outside this interval, points outside of the segment, but still on the line are addressed extrapolation.

Connection with linear functionsEdit

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

  • Additivity:  
  • Homogeneity of degree 1:  

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, when written in this manner, produce functions which will have neither property above and hence are not linear functions in this sense. They are known as affine functions.

ExampleEdit

An everyday example of the use of different forms of linear equations is computation of tax with tax brackets. This is commonly done with a progressive tax computation, using either point–slope form or slope–intercept form.

More than two variablesEdit

For the general case of a linear equation with   unknowns the equation may always be assumed to be denoted as at the top

 

Sometimes   is called the absolute term, and the term coefficient is reserved for the   A variant to denote   stemming from the use in polynomials, is to write   instead, alluding to the zeroth power of any variable, that always reduces to  

When dealing with   variables, it is common to use   and   instead of indexed variables.

The set of solutions of such an equation is a set of  -tuples, and each  -tuple makes the equation an identity, when its components are inserted for the respective unknowns. The values of the variables with zero coefficients are taken arbitrarily from the field of coefficients.

For an equation to have meaningful solutions, at least one coefficient must be non-zero. This can be formulated as

 

If all coefficients   equal zero, then, as mentioned for one variable, the equation is either inconsistent (for  ) and there is no solution, or all  -tuples are solutions.

The set of solutions ( -tuples) of a linear equation in   variables is an  -dimensional hyperplane in an  -dimensional Euclidean space (or affine space if the coefficients are complex numbers or belong to any field). Within the usual setting of real numbers and a three-dimensional space with Cartesian coordinates, the set of the solutions of a linear equation with three variables describes a plane in the intuitive sense.

A given equation may be solved for all variables with a non-zero coefficient. Let   be an index such that   then

 

This way the linear equation can be seen as defining a function of   variables, which maps, assuming the setting of reals, the set of  -tuples[2] of reals to the real numbers, i.e.:

 

See alsoEdit

NotesEdit

  1. ^ Barnett, Ziegler & Byleen 2008, pg. 15
  2. ^ The (n-1)-tuples are ordered to represent the removal of j from the sequence {1..n}.

ReferencesEdit

  • Barnett, R.A.; Ziegler, M.R.; Byleen, K.E. (2008), College Mathematics for Business, Economics, Life Sciences and the Social Sciences (11th ed.), Upper Saddle River, N.J.: Pearson, ISBN 0-13-157225-3

External linksEdit