Question 362348
When your equations lead to an inconsistency, such as, {{{1=0}}} or similar, then you have two parallel lines with no solution.
Example:
1.{{{2x+y=12}}}
2.{{{2x+y=5}}}
Subtract eq. 2 from eq. 1,
{{{2x+y-2x-y=12-5}}}
{{{0=7}}} 
No solution, the lines are parallel.
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When your equations lead to an identity, such as, {{{5=5}}} or similar, then you have two lines that are the same line and have infinitely many solutions.
1.{{{2x+y=6}}}
2.{{{4x+2y=12}}}
Multiply eq. 1 by -2 and add to eq. 1,
{{{-4x-2y+4x+2y=-12+12}}}
{{{0=0}}}
The lines are the same line (eq. 2 is 2 times eq. 1), there are infinitely many solutions (every solution of eq. 1 is also a solution of eq. 2).