For every point $P(x ; y)$ in the $(x ; y)$ coordinate plane, we assign the point $P^{\prime}(x-y ;-y)$. Which are the lines that remain unchanged under this transformation?
Solution. The sought lines either do not change during the transformation, so-called fixed lines; or they map onto themselves, invariant lines.

If the line does not change, it means that every point is identical to its image. Let $P(x ; y), P^{\prime}(x-y ;-y)$, then $x=x-y$, and $y=-y$. Both equalities express that $y=0$. The fixed line of the transformation is therefore the $x$-axis.

If the line maps onto itself during the transformation, this means that the slope of the line and its image are the same (it immediately follows that the image of vertical lines is not vertical, so an invariant line has a slope) and both intersect the $y$-axis at the same point.

Let the equation of $e$ be $y=m x+b$, then the equation of $e^{\prime}$ is $-y=m(x-y)+b$. This will be the equation of the line $y=m x+b$ if and only if $m=2$. Therefore, every line with a slope of 2 is invariant.