Definition:Simultaneous Equations/Linear Equations/Solution

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Definition

Consider the system of simultaneous linear equations:

$\ds \forall i \in \set {1, 2, \ldots, m} : \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$


That is:

\(\ds \beta_1\) \(=\) \(\ds \alpha_{1 1} x_1 + \alpha_{1 2} x_2 + \cdots + \alpha_{1 n} x_n\)
\(\ds \beta_2\) \(=\) \(\ds \alpha_{2 1} x_1 + \alpha_{2 2} x_2 + \cdots + \alpha_{2 n} x_n\)
\(\ds \) \(\cdots\) \(\ds \)
\(\ds \beta_m\) \(=\) \(\ds \alpha_{m 1} x_1 + \alpha_{m 2} x_2 + \cdots + \alpha_{m n} x_n\)


Let $\tuple {x_1, x_2, \ldots, x_n}$ satisfy each of the equations in $\ds \sum_{j \mathop = 1}^n \alpha_{i j} x_j = \beta_i$.

Then $\tuple {x_1, x_2, \ldots, x_n}$ is referred to as a solution to the system of simultaneous linear equations


Also see

  • Results about simultaneous linear equations can be found here.


Sources