[tex]\left\{\begin{array}{ccc}ax+by=c\\ax+by=c\end{array}\right\Rightarrow\text{coincident}\\\\\left\{\begin{array}{ccc}ax+by=c\\ax+by=d\\c\neq d\end{array}\right\Rightarrow\text{inconsistent}\\\\\left\{\begin{array}{ccc}ax+by=c\\dx+ey=c\\a\neq d\ or\ b\neq e\end{array}\right\Rightarrow\text{consistent independent}[/tex]
[tex]\left\{\begin{array}{ccc}x+5y=-2\\x+5y=4\end{array}\right\Rightarrow\boxed{\text{inconsistent}}\\--------------------------------\\\left\{\begin{array}{ccc}y=3x+4&\text{subtract 3x from both sides}\\-2x+y=4\end{array}\right\\\\\left\{\begin{array}{ccc}-3x+y=4\\-2x+y=4\end{array}\right\Rightarrow\boxed{\text{consistent independent}}\\-----------------------------\\\left\{\begin{array}{ccc}3x+y=4\\-6x-2y=-8&\text{divide both sides by (-2)}\end{array}\right\\[/tex]
[tex]\left\{\begin{array}{ccc}3x+y=4\\3x+y=4\end{array}\right\Rightarrow\boxed{\text{coincident}}[/tex]
