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See also: System of Linear Equations
See also: Solving for Two Equations and Two Unknowns
See also: System of Linear Equations, Row Echelon Form
If we have an equation: \(a\cdot x_1 +b\cdot x_2 +c\cdot x_3 =o\), we can write this in matrix form:
This is a linear equation, where we have a sequence of variables multiplied by coefficients, more generally, this is a linear equation with \(n\) unknown variables, and \(n+1\) known coefficients, note the \(a\) at the beginning:
In 2 dimension (with two unknowns), this is a line; in 3 dimension, this is a surface.
We have a system of linear equations, 3 equations and 3 unknowns:
We can define these:
\(\displaystyle W=\left\lbrack \begin{array}{ccc} a & b & c\\ d & e & f\\ g & h & i \end{array}\right\rbrack\)
\(\displaystyle X=\left\lbrack \begin{array}{c} x_1 \\ x_2 \\ x_3 \end{array}\right\rbrack\)
\(\displaystyle v=\left\lbrack \begin{array}{c} o\\ p\\ q \end{array}\right\rbrack\)
We can write \(W\) and \(v\) together like this, this is the augmented matrix of the system of linear equations: