Difference between revisions of "Linear equation"
(New page: Linear equations are any algebraic equations where both sides of the equation are polynomials of the first degree. == single-variable equations == Single-variable linear equation...) |
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| − | + | In [[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. | |
| − | == | + | == Form and Connection to Analytic Geometry == |
| + | 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]]. | ||
| − | + | 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 == | |
| + | === Variable Elimination === | ||
| + | === Matrices and Cramer's Law === | ||
| − | + | ==See Also== | |
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| − | + | [[Category:Algebra]] | |
| − | + | [[Category:Definition]] | |
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Latest revision as of 12:39, 14 July 2021
In 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.
Contents
Form and Connection to Analytic Geometry
In general, a linear equation with 
variables can be written in the form 
, where 
is a series of constants, 
is a series of variables, and 
is a constant.
In other words, a linear equation is an equation that can be written in the form ![\[a_1b_1 + a_2b_2 + ... +a_nb_n = c\]](http://latex.artofproblemsolving.com/c/9/4/c9476b2075c29a42857bff61474da72a716287ef.png)
, where 
are constants multiplied by variables 
and is a constant.
For the particular case 
(single variable equation), the resulting equation can be graphed as a point on the number line, and for the case 
(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.
