Find the relation between $x$ and $y$ if the point $(x, y)$ is equidistant from $(7, 1)$ and $(3, 5)$.

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Step-by-Step Solution

Step 1: Define the points

We are given three points: $P(x, y)$, which is an arbitrary point, and two fixed points, $A(7, 1)$ and $B(3, 5)$. The problem states that point $P$ is equidistant from points $A$ and $B.

Step 2: Apply the distance formula

Since point $P$ is equidistant from $A$ and $B$, the distance $PA$ must be equal to the distance $PB$. To simplify calculations and avoid square roots, we can equate the squares of these distances, $PA^2 = PB^2$. We then apply the distance formula, which states that the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

Step 3: Expand and simplify the equation

We expand the squared terms on both sides of the equation. Remember the algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$. After expansion, we will collect like terms and simplify.

Step 4: Rearrange and solve for the relation

First, we cancel out $x^2$ and $y^2$ from both sides of the equation. Then, we gather all $x$ and $y$ terms on one side and constant terms on the other. Finally, we simplify the equation by dividing by a common factor to find the linear relation between $x$ and $y.