Problem:
Given point \( P(x, y) \) is equally distant from points \( A(7, 1) \) and \( B(3, 5) \).
Determine the relationship between \( x \) and \( y \).
Step 1: Apply the equidistant definition
If \( P \) is equidistant from \( A \) and \( B \), then:
\[PA = PB\]
Using the distance formula:
\[\sqrt{(x - 7)^2 + (y - 1)^2} = \sqrt{(x - 3)^2 + (y - 5)^2}\] Step 2: Square both sides
\[(x - 7)^2 + (y - 1)^2 = (x - 3)^2 + (y - 5)^2\] Step 3: Expand
Left-hand side:\
\[(x - 7)^2 = x^2 - 14x + 49 \\(y - 1)^2 = y^2 - 2y + 1 \\\Rightarrow x^2 - 14x + 49 + y^2 - 2y + 1 = x^2 + y^2 - 14x - 2y + 50\]
Right-hand side:\
\[(x - 3)^2 = x^2 - 6x + 9 \\(y - 5)^2 = y^2 - 10y + 25 \\\Rightarrow x^2 - 6x + 9 + y^2 - 10y + 25 = x^2 + y^2 - 6x - 10y + 34\] Step 4: Cancel common terms
Remove \( x^2 \) and \( y^2 \) from both sides:
\[-14x - 2y + 50 = -6x - 10y + 34\] Step 5: Isolate terms
\[-14x + 6x - 2y + 10y = 34 - 50\Rightarrow -8x + 8y = -16\] Step 6: Simplify
\[-8x + 8y = -16 \Rightarrow x = y + 2\] Final Answer:
The x-coordinate is \( \boxed{2 \text{ more than the y-coordinate}} \), i.e., \( \boxed{x = y + 2} \)\