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 linear 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
- 1982: A.O. Morris: Linear Algebra: An Introduction (2nd ed.) ... (previous) ... (next): Chapter $1$: Linear Equations and Matrices: $1.3$ Applications to Linear Equations: Definition $1.6$