Step 1: Let the required point be $P(x, y)$ lying on $x + y = 5$, so $y = 5 - x$.
Thus $P(x,\ 5-x)$.
Step 2: Given $PA = PB$ where $A(6,4)$ and $B(5,2)$. Square both sides :
$$PA^2 = PB^2$$
Step 3: $$(x-6)^2 + (5-x-4)^2 = (x-5)^2 + (5-x-2)^2$$
$$(x-6)^2 + (1-x)^2 = (x-5)^2 + (3-x)^2$$
Step 4: Expand.
$$x^2 - 12x + 36 + 1 - 2x + x^2 = x^2 - 10x + 25 + 9 - 6x + x^2$$
$$-14x + 37 = -16x + 34$$
Step 5: $$2x = -3 \Rightarrow x = \frac{-3}{2}$$
Step 6: $$y = 5 - \left(\frac{-3}{2}\right) = \frac{13}{2}$$
Hence the required point is $\left(\dfrac{-3}{2},\ \dfrac{13}{2}\right)$.