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	PlanetMath: finite-dimensional linear problem
		The equation is called under-determined or over-determined depending on whether the former is greater than the latter, or vice versa.
		This is unfortunate, because the concept of a linear equation is much more general than the traditional notion of “variables and equations”, and relies in an essential way on the idea of a linear mapping.
		Polynomial interpolation is a linear problem, but one that is specified abstractly, rather than in terms of variables and equations.
	planetmath.org /encyclopedia/SystemOfLinearEquations.html (299 words)

	PlanetMath: linear equation
		The equation is called inconsistent if no solutions exist, that is, if the pre-image is the empty set.
		They neglect to mention the underlying mapping, preferring to focus instead on “variables and equations.” However, the scope of the general concept is considerably wider, e.g.
		This is version 4 of linear equation, born on 2002-02-22, modified 2006-06-15.
	planetmath.org /encyclopedia/LinearProblem.html (167 words)

	Linear equation - Wikipedia, the free encyclopedia
		A linear equation is an equation involving only the sum of constants or products of constants and the first power of a variable.
		Such an equation is equivalent to equating a first-degree polynomial to zero.
		Because of the linear property above, the solutions of linear equations of this kind can in general be described as a superposition of other solutions of the same equation.
	en.wikipedia.org /wiki/Linear_equation (913 words)

	MathSteps: Grade 7: Linear Equations: What Is It?
		No variable in a linear equation is raised to a power greater than 1 or used as the denominator of a fraction.
		A linear equation in two variables describes a relationship in which the value of one of the variables depends on the value of the other variable.
		The real importance of emphasizing graphing linear equations with your students, is that they should already know that any two points determine a line, so finding many pairs of values that satisfy a linear equation is easy: Find two pairs of values and draw a line through the points they describe.
	www.eduplace.com /math/mathsteps/7/d/index.html (1906 words)

	Linear Equations
		Linear equations are equations involving only one variable, like x or y, and nothing complicated like powers or square roots.
		Linear equations are equations involving only one variable, like x or y, and they do not involve anything complicated like powers, square roots, or anything like that.
		The reason for this is that linear equations are the first opportunity to practice operating with equations and develop equation solving skills.
	www.valgetal.com /linear_equations.htm (641 words)

	Linear Equation (Site not responding. Last check: 2007-11-06)
		Line equation in the standard form is written as ax + by = c (where a, b, and c are numerals, and x and y are variables).
		Line equation in the slope-intercept form is written as y = mx + b (where m and b are numerals, and x and y are variables) in which m is the slope and b is the y-intercept.
		Write an equation of the line with the slope of 5 passing through the point (-2, -11).
	www.gomath.com /htdocs/lesson/linearequation_lesson1.htm (406 words)

	SparkNotes: Writing Equations: General Linear Form
		General linear form is not the most useful form to use when writing an equation from a graph.
		However, the form highlights certain abstract properties of linear equations, and you may be asked to put other linear equations into this form.
		This equation is linear and the two intercept points satisfy it, therefore it represents the line.
	www.sparknotes.com /math/algebra1/writingequations/section3.rhtml (471 words)

	Linear equations - A complete course in algebra
		That is, formally, to solve an equation we must isolate x on the left of the equal sign.
		The standard form of a linear equation is ax + b = 0, not 0 = ax + b.
		This equation, incidentally, is in the standard form of a linear equation, namely, ax + b = 0.
	www.themathpage.com /alg/equations.htm (1122 words)

	Linear differential equation - Wikipedia, the free encyclopedia
		The linearity condition on L rules out operations such as taking the square of the derivative of y; but permits, for example, taking the second derivative of y.
		If y is assumed to be a function of only one variable, one speaks about an ordinary differential equation, else the derivatives and their coefficients must be understood as (contracted) vectors, matrices or tensors of higher rank, and we have a (linear) partial differential equation.
		(x) by either the method of undetermined coefficients or the method of variation of parameters; the general solution to the linear differential equation is the sum of the general solution of the related homogeneous equation and the particular solution.
	en.wikipedia.org /wiki/Linear_differential_equation (821 words)

	BioMath: Linear Functions
		Here, we discuss solving linear equations beginning with a linear equation in one variable, and then solving a system of two linear equations by two different methods.
		Solving linear equations in one variable is straightforward, as illustrated by the following example.
		Solving a system of two linear equations is the equivalent of finding the point (x, y) where the two lines intersect.
	www.biology.arizona.edu /biomath/tutorials/Linear/LinearEquations.html (1220 words)

	Nonlinearity - Wikipedia, the free encyclopedia (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-11-06)
		In that case if the linear function is continuous, homogeneity is not an additional axiom to establish if the additivity property is established.) Important examples of linear operators include the derivative considered as a differential operator, and many constructed from it, such as del and the Laplacian.
		Equations involving maps include linear equations, and nonlinear equations as well as nonlinear systems (the last is a misnomer stemming from matrix equation 'systems', a nonlinear equation can be a scalar valued or matrix valued equation).
		For nonlinear equations the solutions to the equations do not in general form a vector space and cannot (in general) be superposed (added together) to produce new solutions.
	en.wikipedia.org.cob-web.org:8888 /wiki/Non-linear (844 words)

	linearequations
		An equation is referred to as a linear equation, if the variable we have to solve for has no powers and does not appear in the denominator of a fraction.
		In solving any equation our goal is to find the value that could replace the variable in such a way that the equation remains true.
		Now since this is an equation and we are allowed to perform any arithmetic operation as long as we do it on both sides, we can attempt to eliminate the denominators.
	www.csun.edu /~ayk38384/notes/mod10/linearequations.html (990 words)

	Grade 9: Linear Equations 1
		learn to solve and verify first-degree, single variable equations of the form ax + b = cx + d where a, b, c, and d are rational numbers.
		Linear Equation - move the sliders and study the graph pattern
		After entering an equation of the form ax + b = cx + d, click "Solve" to see the steps and solution Change the coefficients and constants and see the changes in the steps required to reach the solution.
	argyll.epsb.ca /jreed/math9/strand2/2202.htm (521 words)

	Systems of Linear Equations: Gaussian Elimination
		There are numerical techniques which help to approximate nonlinear systems with linear ones in the hope that the solutions of the linear systems are close enough to the solutions of the nonlinear systems.
		A linear system is a set of linear equations and a homogeneous linear system is a set of homogeneous linear equations.
		In general if the linear system has n equations with m unknowns, then the matrix coefficient will be a nxm matrix and the augmented matrix an nx(m+1) matrix.
	www.sosmath.com /matrix/system1/system1.html (1054 words)

	WebQuest
		When you have an equation for a line (linear relation) it can be used in reverse.
		You have the equation of a line and the coordinates (X,Y) for one point that is not on that line.
		Part II Explain in a few paragraphs one example of how a linear equation (relations) is used in everyday use.
	www.gmpdc.org /webquests/sgiguere/linearequations (896 words)

	Second Order Linear Equations
		The theory of linear differential equations is closely related to the theory of systems of linear equations in linear algebra.
		This is a linear transformation in the sense of linear algebra, but the ``vectors'' it operates on are functions.
		For a set of two functions, linear independence is easy to check: they are linearly independent unless one is a constant multiple of the other.
	www.math.ubc.ca /~israel/m215/lin2o/lin2o.html (653 words)

	Math Forum - Ask Dr. Math Archives: High School Linear Equations
		Given the equation 2x+5y=8, with the coefficients increasing by 3 each time, create linear systems that are similar using equations such as 3x+4y=5 or 4x+6y=8.
		Determine whether the graph of each equation rises to the right or falls to the right, is horizontal, or is vertical.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /library/drmath/sets/high_lineareq.html (1049 words)

	Solving One-Step Linear Equations
		Linear equations are the simplest equations that you'll deal with.
		However, since you're older now, the equations can be much more complicated, and therefore the methods you'll use to solve the equations will be a bit more advanced.
		In general, to solve an equation for a given variable, you need to "undo" whatever has been done to the variable.
	www.purplemath.com /modules/solvelin.htm (493 words)

	Equations of Straight Lines (via CobWeb/3.1 planetlab2.cs.umd.edu) (Site not responding. Last check: 2007-11-06)
		When working with straight lines, there are often many ways to arrive at an equation or a graph.
		This equation also describes what is happening to the y-coordinates on the line.
		Substitute the x-values into the equation to determine the y-values.
	regentsprep.org.cob-web.org:8888 /Regents/math/glines/EqLines.htm (290 words)

	Grade 7: Linear Equations 2
		illustrate the solution process for a one-step, single-variable, first-degree linear equation, using concrete materials or diagrams to solve multiplication and division questions.
		explain how to solve simpler multiplication and division linear equation problems, using informal algebraic methods.
		solve and verify one-step multiplication and division linear equations, using a variety of techniques.
	argyll.epsb.ca /jreed/math7/strand2/2203.htm (388 words)

	Straight-Line Equations: Slope-Intercept Form
		There are different types of "standard" formats for straight lines; the "standard" format your book refers to may differ from that used in some other books.
		These "standard" forms are often holdovers from a few centuries ago, when mathematicians couldn't handle very complicated equations, so they tended to obsess about the simple cases.
		You will be given problems where they give you some pieces of information about a line, and they want you to come up with the equation of the line.
	www.purplemath.com /modules/strtlneq.htm (428 words)

	Series Solution of Non-Linear Equation (via CobWeb/3.1 planetlab2.cs.umd.edu) (Site not responding. Last check: 2007-11-06)
		The coefficients of the power series solutions of certain non-linear differential equations are generated by convolutions of the preceeding coefficients.
		One example is the differential equation x x'' + a (x')^2 = b (1) Among the solutions of this equation (with appropriate choices of a,b) are exp(t), sin(t), cos(t), (A+Bt)^n, A+Bt+Ct^2, and sqrt(A+Bt+Ct^2).
		etc Incidentally, the value of b in the ubiquitous equation (1) is essentially just a constant of integration, and the underlying relation is the derivitive x x'' + q x' x'' = 0 where q=3 for unaccelerated separations and q=2 for (non-rotating) gravitational separations.
	www.mathpages.com.cob-web.org:8888 /home/kmath215.htm (316 words)

	Linear Homogeneous Ordinary Differential Equations with Constant Coefficients
		Linear Homogeneous Ordinary Differential Equations with Constant Coefficients
		A linear homogeneous ordinary differential equation with constant coefficients has the general form of
		This equation implies that the solution is a function whose derivatives keep the same form as the function itself and do not explicitly contain the independent variable
	www.efunda.com /math/ode/linearode_consthomo.cfm (170 words)

	Math 1010 on-line - Linear Equations
		It is not possible, for example, that a linear equation has two solutions.
		The crux of solving a linear equation is to recognize that the equation is linear, and to convert it to the above simple form.
		Below you will see several examples of how an equation whose linearity is not obvious can be converted to a clearly linear equation.
	www.math.utah.edu /online/1010/linear/index.html (453 words)

	Linear Equation -- from Wolfram MathWorld
		A linear equation is an algebraic equation of the form
		are linear equations of a single variable, and
		is an example of a linear equation with three variables.
	mathworld.wolfram.com /LinearEquation.html (107 words)

	Question on Defining Linear Equation
		In order to have a linear transformation, you must have some vector space to give you the vectors you apply the linear transformation.
		Any linear transformation must be "over" some number field- the rational numbers, the real numbers, the complex numbers.
		A polyonomial p which satisfies p(x) = ax + b for some constants a, b is (often) called linear (or "a linear polynomial"), probably because its graph resembles a line.
	www.physicsforums.com /showthread.php?t=89943 (1130 words)

	Linear Equations Worksheets, Systems of Linear Equations Worksheets, and Inequalities Worksheets.
		Linear Equations - State whether the line is parallel or perpendicular
		Inequalities - Sketch the graph of the inequality
		Linear Systems - Solve by using the substitution method (in Ax + By = C format)
	www.edhelper.com /LinearEquations.htm (569 words)

	snlp
		The student will investigate the relationship between fat grams and calories, and calculate an equations for approximating these values.
		Using the graph, determine the slope and y-intercept, then write a linear equation for the data
		Compare the equation created manually, and by the graphing calculator, write a journal entry explaining any variations between the two.
	www.selu.edu /Grants/LASIP/lessons/snlp.htm (361 words)

	spuc: Solving Linear Equation Systems
		Solve linear equation system, when LU-factorisation is given.
		Solve linear (band) equation system, when LU-factorisation is given.
		Hence, the equations A*xj=bj is solved for all j=1..p.
	spuc.sourceforge.net /group__linearequations.html (596 words)

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