The points $(5,3)$ and $(1,-1)$ are plotted on a sheet of graph paper. The sheet of graph paper is folded along a line so that the point $(5,3)$ lands on top of the point $(1,-1)$. The equation of the line that represents the fold is
(A) $y=-x+1$
(D) $y=-x+4$
(B) $y=-x+2$
(E) $y=-x+5$
(C) $y=-x+3$
Since the point $(5,3)$ lands on the point $(1,-1)$ when folded, then the fold line must pass through the midpoint of these two points, namely $\left(\frac{1}{2}(5+1), \frac{1}{2}(3+(-1))\right)=(3,1)$.

![](https://cdn.mathpix.com/cropped/2024_04_20_33a8a6559d675f59baa2g-207.jpg?height=547&width=499&top_left_y=1339&top_left_x=867)

Of the given possibilities, $(3,1)$ lies only on the line $y=-x+4$, so (D) is the answer.

(In fact, the fold line must be the perpendicular bisector of the line segment through $(5,3)$ and $(1,-1)$. The slope of the line segment through $(5,3)$ and $(1,-1)$ is $\frac{3-(-1)}{5-1}=1$, so the perpendicular bisector has slope -1 .

Since the perpendicular bisector has slope -1 and passes through $(3,1)$, then it has equation $y=-x+4$.)