Think geometrically,for a change.

The equations correspond to equations of a line. So searching for simultaneous solutions is searching for intersections geometrically.

When will two straight lines in a plane never meet?

When will two straight lines in a plane meet at more than one point?

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Of course, you can do this using linear algebra too.

implies there exists "no simultaneous solution" or "more than one simultaneous solution."

So

,

If

then equations are identical and we have "more than one simultaneous solution."

else

we have "no simultaneous solution".