Distinct Integer Solutions for the Equation ( frac{3x}{x-1} y - x 1 ) and Its Graphical Interpretation

Consider the equation (frac{3x}{x-1} y - x 1). This equation represents a relationship between integers (x) and (y). We aim to determine the number of distinct integer solutions for this equation and provide a graphical interpretation of the solutions.

Mathematical Analysis

First, we simplify and analyze the given equation:

Given: (frac{3x}{x-1} y - x 1)

Let's start by simplifying the left side:

[ frac{3x}{x-1} frac{3x - 3 3}{x-1} 3 frac{3}{x-1} ]

Substituting this into the equation:

[ 3 frac{3}{x-1} y - x 1 ]

Rearranging to solve for (y):

[ y x - 2 frac{3}{x-1} ]

For (y) to be an integer, (frac{3}{x-1}) must be an integer. This implies that (x-1) is a divisor of 3.

Integer Solutions

The divisors of 3 are (pm1) and (pm3). Therefore, we have:

(x-1 1 Rightarrow x 2) (x-1 -1 Rightarrow x 0) (x-1 3 Rightarrow x 4) (x-1 -3 Rightarrow x -2)

For each of these integer values, we can determine the corresponding (y) values:

For (x 2):

( y 2 - 2 frac{3}{2-1} 3 )

Thus, ((x, y) (2, 3)).

For (x 0):

( y 0 - 2 frac{3}{0-1} 0 - 2 - 3 -5 )

Thus, ((x, y) (0, -5)).

For (x 4):

( y 4 - 2 frac{3}{4-1} 2 1 3 )

Thus, ((x, y) (4, 1)).

For (x -2):

( y -2 - 2 frac{3}{-2-1} -4 - 1 -5 )

Thus, ((x, y) (-2, -5)).

Summarizing the distinct integer solutions, we have:

(2, 3) (0, -5) (4, 1) (-2, -5)

Hence, the number of distinct integer solutions is boxed{4}.

Graphical Interpretation

The given equation can be interpreted graphically by examining the behavior of the function (y x - 2 frac{3}{x-1}). The function has a removable discontinuity at (x 1), and the values of (y) are determined by the divisors of 3 as analyzed.

The intersection points with integer values for (x) and (y) provide distinct integer solutions. The graphical representation confirms these solutions, giving a visual interpretation of the equation's behavior.