To find a relationship between $x$ and $y$ such that the point $(x, y)$ is equidistant from the two given points $(4, -1)$ and $(-3, 5)$, we can use the concept that the distance between two points in the plane is given by the distance formula.

Step 1: Distance Formula

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step 2: Apply the Distance Formula

Let $(x, y)$ be the point that is equidistant from $(4, -1)$ and $(-3, 5)$. This means the distance from $(x, y)$ to $(4, -1)$ must equal the distance from $(x, y)$ to $(-3, 5)$. Thus, we have the following equation:

$\sqrt{(x - 4)^2 + (y + 1)^2} = \sqrt{(x + 3)^2 + (y - 5)^2}$

Step 3: Eliminate the Square Roots

To eliminate the square roots, square both sides of the equation:

$(x - 4)^2 + (y + 1)^2 = (x + 3)^2 + (y - 5)^2$

Step 4: Expand Both Sides

Now, expand both sides of the equation:

$(x^2 - 8x + 16) + (y^2 + 2y + 1) = (x^2 + 6x + 9) + (y^2 - 10y + 25)$

Step 5: Simplify the Equation

Cancel out the $x^2$ and $y^2$ terms (since they appear on both sides):

$-8x + 16 + 2y + 1 = 6x + 9 - 10y + 25$

Combine like terms:

$-8x + 2y + 17 = 6x - 10y + 34$

Step 6: Rearrange Terms

Move all terms involving $x$ and $y$ to one side:

$-8x - 6x + 2y + 10y = 34 - 17$

Simplify:

$-14x + 12y = 17$

Step 7: Final Equation

Divide the entire equation by 2 to simplify it:

$-7x + 6y = \frac{17}{2}$

Thus, the relationship between $x$ and $y$ such that the point $(x, y)$ is equidistant from the points $(4, -1)$ and $(-3, 5)$ is:

$-7x + 6y = \frac{17}{2}$

This is the equation of the perpendicular bisector of the line segment joining the two points.

Would you like more details or clarification on any step?

Here are 5 related questions to explore:

1. How do you find the midpoint between two points?
2. What is the equation of a circle passing through points $(4, -1)$ and $(-3, 5)$?
3. How can you find the slope of the line segment joining two points?
4. What is the geometric significance of the perpendicular bisector?
5. How do you calculate the distance between two points in 3D space?

Tip: The perpendicular bisector is always equidistant from two points—useful for constructing geometric shapes like triangles.