Prove $1/x + 1/y = 0$ is not linear.

Question

I was trying to complete an exercise of a book and tried to solve this question but wasn't able to succeed. I searched it on google but found no results related to this question. Please help.

Answer 1

If $\dfrac1x+\dfrac1y=0$ then $\dfrac1x=-\dfrac1y$ so $y=-x$, which is of the linear form $y=mx+b$ with $m=-1$ and $b=0$, but $x\ne0$ and $y\ne0$ (because if $z=0$ then $\dfrac1z$ is undefined). So the locus of points $(x,y)$ satisfying $\dfrac1x+\dfrac1y=0$ is a line with a point (the origin) missing.