Difference between revisions of "Linear equation"

Revision as of 20:15, 4 December 2008

In elementary algebra, linear equations are algebraic equations in which both sides of the equation are polynomials or monomials of the first degree - i.e. each term does not have any variables to a power other than one.

Form and Connection to Analytic Geometry

In general, a linear equation with $n$ variables can be written in the form $\displaystyle\sum_{i=1}^{n}a_ib_i=c$ , where $a_i$ is a series of constants, $b_i$ is a series of variables, and $c$ is a constant.

For the particular case $n=1$ , the resulting equation can be graphed as a point on the number line, and for the case $n=2$ (resulting in a linear function), it can be graphed as a line on the Cartesian plane, hence the term "linear" equation. This can extended to a general Cartesian n-space, in which the linear equation with the corresponding number of variables can be graphed as an n-1-space - this concept is the idea behind analytic geometry as envisioned by Fermat and Descartes.

@@ Line 1: / Line 1: @@
-Linear equations are any algebraic equations where both sides of the equation are [[polynomials]] or [[monomials]] of the first [[degree]].
+In [[elementary algebra]], '''linear equations''' are algebraic [[equation]]s in which both sides of the equation are [[polynomials]] or [[monomials]] of the first [[degree]] - i.e. each term does not have any variables to a power other than one.
-== Single-Variable Equations ==
+== Form and Connection to Analytic Geometry ==
+In general, a linear equation with <math>n</math> variables can be written in the form <math>\displaystyle\sum_{i=1}^{n}a_ib_i=c</math>, where <math>a_i</math> is a series of constants, <math>b_i</math> is a series of variables, and <math>c</math> is a constant.
-Single-variable linear equations can '''always''' be expressed in the form:
+For the particular case <math>n=1</math>, the resulting equation can be graphed as a point on the number line, and for the case <math>n=2</math> (resulting in a [[linear function]]), it can be graphed as a line on the [[Cartesian plane]], hence the term "linear" equation. This can extended to a general Cartesian n-space, in which the linear equation with the corresponding number of variables can be graphed as an n-1-space - this concept is the idea behind [[analytic geometry]] as envisioned by Fermat and Descartes.
-<math>ax+b=0</math>
-where <math>a</math> and <math>b</math> are constants and <math>x</math> is the variable.
+== Systems, solutions and methods of solving ==
+=== Variable Elimination ===
+=== Matrices and Cramer's Law ===
-From this form, they can be solved with the following steps:
+==See Also==
-. subtract <math>b</math> from both sides.
+[[Category:Analytic geometry]]]
+[[Category:Definition]]
-. divide <math>a</math> from both sides.
+[[Category:Elementary algebra]]
-== Two-Variable Equations ==
-Two-variable linear equations can '''always''' be expressed in the form:
-<math>y=mx+b</math>
-Where <math>y</math> and <math>x</math> are variables and <math>m</math> and <math>b</math> are constants.
-Two-variable linear equations can be drawn as a straight line on a coordinate plane.
-<math>m</math> can be defined as the [[slope]] and <math>b</math> can be defined as the [[y-intercept]]

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Difference between revisions of "Linear equation"

Revision as of 20:15, 4 December 2008

Contents

Form and Connection to Analytic Geometry

Systems, solutions and methods of solving

Variable Elimination

Matrices and Cramer's Law

See Also