Difference between revisions of "Linear equation"

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== Form and Connection to Analytic Geometry ==
 
== 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.  
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In general, a linear equation with <math>n</math> variables can be written in the form <math>\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]].  
  
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.
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In other words, a linear equation is an equation that can be written in the form <cmath>a_1b_1 + a_2b_2 + ... +a_nb_n = c</cmath>, where <math>a_1, a_2,... , a_n</math> are constants multiplied by variables <math>b_1, b_2, ..., b_n</math> and <math>c</math> is a constant.
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For the particular case <math>n=1</math> (single variable equation), 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.
  
 
== Systems, solutions and methods of solving ==  
 
== Systems, solutions and methods of solving ==  
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==See Also==
 
==See Also==
  
[[Category:Analytic geometry]]
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[[Category:Definition]]
 
[[Category:Definition]]
 
[[Category:Elementary algebra]]
 
[[Category:Elementary algebra]]

Revision as of 20:11, 23 January 2017

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 $\sum_{i=1}^{n}a_ib_i=c$, where $a_i$ is a series of constants, $b_i$ is a series of variables, and $c$ is a constant.

In other words, a linear equation is an equation that can be written in the form \[a_1b_1 + a_2b_2 + ... +a_nb_n = c\], where $a_1, a_2,... , a_n$ are constants multiplied by variables $b_1, b_2, ..., b_n$ and $c$ is a constant.

For the particular case $n=1$ (single variable equation), 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.

Systems, solutions and methods of solving

Variable Elimination

Matrices and Cramer's Law

See Also