It's simpler than that.Originally Posted byBoban

These equations represent a pair if lines in the plane.

They have no solutions when the lines are parallel, but not the same line;

they have a single unique solution if they are not parallel, and an infinite

number of solutions if they represent the same line.

These equations represent the same line when they are a multiple of

one another.

If

Then the RHS of the second equation is times the RHS of the

first equation, so the same must be true of the LHS. So multiplying

the first equation by 3 and then equating coefficients of and

, which gives:

,

.

Which is a pair of linear equations which can be solved for and

to give and which give an infinite number of solutions

to the original equations.

If

Both equations reduce to , so there are an infinite

number of solutions (that is any point on the x-axis is a solution).

If

The equations become:

,

which implies that .

If

Left as an exercise to the reader.

RonL