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System of Linear Equations

There are systems of equations with certain solution.
For example, if we have a system of equation as following:
$\begin{cases} 3x + 5y + z = 2\\ 7x + 2y + 4z = 9 \\ -6x + 3y + 2z = 1\end{cases}$
If you look at it carefully, these equations look like scalar equation of planes. The solution to this system of equation can be interpreted as the point that is in al three planes, or rather the intersection of the planes.
The textbook explains system of equation and elimination a lot better than I can paraphrase. Therefore, if you want to learn more about these, please refer to the textbook.

0.1. System of Equation in Matrix Form System of equation can be expressed as an augmented matrix. This matrix is in the form of $[A|\vec{b}]$ where A is a matrix formed by the coefficients of the linear equations, and $\vec{b}$ is the vector made with the “answers” of of each equations.
Putting the above linear equation into an augmented matrix, we get
$
\begin{amatrix}{3}
3 & 5 & 1 &2 \\ 7 & -2 &4 &9\\-6&3&2&1\\
\end{amatrix}
$

0.1.1. Homogeneous System If the system all equals to zero, meaning $\vec{b}$ is the zero vector, then the system is called Homogeneous System. $[A|\vec{0}]$.

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