If point (x, y) lies on the line joining the points (3, 5) and (−2, −3), then the relation between x and y is given by the algebraic relation

If the point (x, y) lies on the line joining the points (3, 5) and (−2, −3), then the area of the triangle formed by these vertices is zero.

The area of a triangle whose vertices are (x₁, y₁), (x₂, y₂), and (x₃, y₃) is given by the numerical value of the expression $\frac{1}{2}\left[x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right]$.

$\therefore \frac{1}{2}\left[x\left(5-(-3)\right)+3\left(-3-y\right)+\left(-2\right)\left(y-5\right)\right]=0$

$\Rightarrow 8x-9-3y-2y+10=0$

$\Rightarrow 8x-5y+1=0$

Thus, the relation between x and y is given by the algebraic relation 8x − 5y + 1 = 0.