Analytical geometry - Finding the coordinates of point M

Question

I've been practicing analytical geometry lately and I've come to a problem. I solved the problem a few times but I can't get the right result.

Here is the math problem:

Point M whose distance from the line (I don't know if it is called line in English) 2x + y - 3 = 0 is $\sqrt{5}$, is also equally distant from points A(4, -3), B(2, -1). Find the coordinates of point M

The correct answers in my book are $M$($\frac{13}{3}$, -$\frac{2}{3}$) and $M$($1$, $-4$)

But this is the method I used and got a completely different answer

First I did this:

$\sqrt{5}$ = $\frac{|2Xm + Ym - 3|}{\sqrt{5}}$ and then I got two equations $2Xm + Ym = 4$ and the second one $2Xm + Ym = 2$

The next thing I did was this:

$\sqrt{(Xm-4)^{2} + (Ym +3)^{2}} =\sqrt{(Xm-2)^{2} + (Ym +1)^{2}} $

Then when I made this equation a little simpler I've gotten:

$Xm - Ym = 5$

Then I made a system with two equations with two variables twice:

First I solved $2Xm + Ym = 4$ and $Xm - Ym = 5$ and I get the answer $M(3, -2)$

Then I solved $2Xm + Ym = 2 $ and $Xm - Ym = 5$ and I get the answer $M(\frac{7}{3},-\frac{9}{3})$

Where did I go wrong ?

Answer 1

Where did I go wrong ?

In the following part :

$\sqrt{5}$ = $\frac{|2Xm + Ym - 3|}{\sqrt{5}}$ and then I got two equations $2Xm + Ym = 4$ and the second one $2Xm + Ym = 2$

It should be the following : $$|2X_m+Y_m-3|=5\implies 2X_m+Y_m-3=\pm 5\implies 2X_m+Y_m=3\pm 5=8, -2$$