Can a system of two linear equations have exactly two solutions?

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Solution

Find out whether a system of linear equation has exactly 2 solutions:
A system of $2$ linear equations can have no solution, one solution or infinitely many solution.
Let the two equations be
$a x_{1} + b y_{1} = c_{1} a x_{2} + b y_{2} = c_{2}$
If $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$ then the system is said to be inconsistent and has no solution.
if $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ then the system is dependent and has infinitely many solutions.
if $\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}$ then the system is consistent and has a unique solution.
Therefore the system of $2$ linear equation will not have $2$ solutions.

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