What is a consistent linear system?

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Explanation for a consistent linear system:
A system of linear equations is called a consistent linear system if at least one solution set exists that satisfies all linear equations.
Consider, $\{\begin{cases} a_{1} x + b_{1} y = c_{1} \\ a_{2} x + b_{2} y = c_{2} \end{cases}$ as a system of two linear equations, where $a_{1}, a_{2}, b_{1}, b_{2}, c_{1}$ and $c_{2}$ are constants; $x$ and $y$ are variables.
Then, $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ and $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ represent a consistent linear system; $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ represent an inconsistent linear system.
The three cases for a system of linear equations:
Case $1$ : $\{\begin{cases} x + y = 3 \\ x - y = 1 \end{cases}$
Here, $a_{1} = 1, a_{2} = 1 and b_{1} = 1, b_{2} = - 1$
It is clear that,
$\frac{a_{1}}{a_{2}} = \frac{1}{1}, \frac{b_{1}}{b_{2}} = \frac{1}{- 1} \Rightarrow \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ .
Hence, the system of the linear equations is consistent and has a unique solution.
Hence intersected lines are represented by the system of these two linear equations.
Case $2$ : $\{\begin{cases} x + y = 3 \\ 2 x + 2 y = 6 \end{cases}$
Here,
$a_{1} = 1, a_{2} = 2 b_{1} = 1, b_{2} = 2 a n d c_{1} = 3, c_{2} = 6$
It is clear that $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ .
Hence, the system of linear equations is consistent and has infinitely many solutions.
Hence, two coincided lines are represented by the system of these two linear equations.
Case $3$ : $\{\begin{cases} x + y = 3 \\ x + y = 4 \end{cases}$
Here,
$a_{1} = 1, a_{2} = 1 b_{1} = 1, b_{2} = 1 a n d c_{1} = 3, c_{2} = 4$
It is clear that $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ .
Hence, the system of linear equations is not a consistent linear system as it has no solution.
Hence, two parallel lines are represented by the system of these two linear equations.
The first two cases represent a consistent linear system and the last case represents an inconsistent linear system.
Therefore, a consistent linear system is one that has at least one solution set satisfying all linear equations in the system.

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