Question

How many solutions does the system formed by x-y=3 and ay-ax=0 have for a nonzero number a?

Answer 1

We have no solution.

Explanation:

For a non-zero `a`, dividng both sides of the equation `ay-ax=0`, it becomes

`y-x=0`.

As `x-y=3`, we have the other equation as `y-x=-3`,

which means `-3=0`. But this is not possible.

Hence, we have no solution.

Note : In such cases, in Cartesian plane, the equations `x-y=3` and `ay-ax=0`, where `a!=0`, represent a pair of lines. As they do not meet, we have no solution.