<aside> 💡 A linear equation is an equation of the form: $a_1x_1+a_2x_2+...+a_nx_n=b$,where $x_1,x_2,..,x_n$ are variables, $a_1,a_2,...,a_n$ are the coefficients, which are real numbers.
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e.g.: $x+2y+z=10$, here $x,y,z$ are the variables and $1,2,1$ are the coefficients.
<aside> 💡 A system of linear equations is a collection of one or more linear equations involving some set of variables.
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e.g.: $7x+2y=24$
$8x+2y=30$
is a system of linear equation in 2 variables $x$and $y$.
<aside> 💡 The System of linear equations is equivalent to a matrix equation of the form: $AX=b$, where $A$ is an $m\times n$ matrix, $X$ is a column matrix with $n$ entries and $b$ is a column matrix with $m$ entries.
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e.g.:
$$ \begin{alignat}{2} 3x+2y+z=2\\ x-\frac 1 2 y+\frac 2 3 z=4 \\4x+6y-10z=0 \end{alignat} $$
can be represented as $AX=b$, where
$$ A= \begin{bmatrix} 3 & 2 & 1 \\ 1 & -\frac1 2 & \frac 2 3\\ 4 & 6 & -10 \end{bmatrix}, x= \begin{bmatrix} x \\ y \\ z \end{bmatrix}, b= \begin{bmatrix} 6 \\ \frac 7 6 \\ 0 \end{bmatrix} $$
There are 3 possibilities for the solutions to a linear system of equations: