<s>[INST] How to find x and y coordinates based on the given distance? The problem says:  Find the point (coordinates $(x,y)=~?$) which is symmetrical to the point $(4,-2) $ considering the given equation $y=2x-3$
I have found the perpendicular line-slope $y=-~\frac{1}{2}x$ and the 
intersection point which is shown in the graph $\left(\frac{6}{5},\frac{-3}{5}\right)$
I'm somehow unable to find the $x$ and $y$ I have found the distance based on the point and point distance equation:$$d(p_1,p_2)= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}=\sqrt{\left(\frac{6}{5}-4\right)^2+\left(-\frac{3}{5}+2\right)^2}= \frac{7\sqrt{5}}{5}$$
$$d_1=d_2=d(p_1,p_2)$$
So now I know the distance, the min. distance to the unknown point is the same.
What is the easiest way to find the symmetrical $x$ and $y$ coordinates?
(This is a high school problem)

 [/INST] Alternative rapid way: calling $x_0$ and $y_0$ the coordinates of the symmetric point, the simplest method is to note that the differences in the $x $- and $y $- coordinates  between the searched point and $( \frac{6}{5}, -\frac{3}{5})$ must equal those between this last point and $(4,-2) $. Thus:
$$ 4-\frac{6}{5} =  \frac{6}{5} -x_0$$
$$ y_0 - \left(-\frac{3}{5}\right) =  -\frac{3}{5} - \left(-2\right) $$
Now solve these equations to find the coordinates.
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