User:Ascii/Definition:Linear Equation
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Definition
A linear equation is a polynomial equation of the form:
- $a x + b = 0$
such that $a \ne 0$.
General Definition
A linear equation is an equation in the form:
- $b = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n$
where all of $a_1, \ldots, a_n, x_1, \ldots x_n, b$ are elements of a given field.
The point is that all the indices of the $x$ and $y$ terms in such an equation are $1$.
= Notes = * Presenting a linear equation as of the form ax = b as an alternative. == Example == === Example 1 === 3x = 15 This is solvable over N. === Example 2 === 7x + 49 = 0 This is solvable over Z. It is not solvable over N, however, and this may be mentioned as a motivation for Z or negative numbers. Possible mention of Diophantine equations here. === Example 3 === 3x + 11 = 0 This is solvable over Q Similar things could be said about it being a motivation for extending Z. === Example 4 === 3x + 4y = 18 A lot of commentary could be made here: Finite solutions over N. Infinite solutions over Z with a mention of linear combinations and ways to generate them. Similar for Q.
Also see