Solving for Integral Solutions in Linear Equations: The Case of 2x - 3y 763

The equation 2x - 3y 763 is an example of a linear Diophantine equation, a type of equation where we seek integer solutions. This article explains the steps to find the number of integral solutions to such an equation. Let's proceed step-by-step to understand and solve the given equation.

Step 1: Rearranging the Equation

To express y in terms of x, we rearrange the given equation:

Rearrange 2x - 3y 763:

3y 2x - 763

Thus, y (frac{2x - 763}{3})

Step 2: Condition for y to be an Integer

For y to be an integer, the numerator (2x - 763) must be divisible by 3. This leads us to the condition:

2x - 763 ≡ 0 (mod 3)

First, we need to find the remainder of 763 when divided by 3:

763 ÷ 3 254 remainder 1, thus 763 ≡ 1 (mod 3)

2x - 763 ≡ 0 (mod 3) simplifies to:

2x - 1 ≡ 0 (mod 3)

2x ≡ 1 (mod 3)

To solve 2x ≡ 1 (mod 3), we test values of x (mod 3):

If x ≡ 0 (mod 3), then 2x ≡ 0 (mod 3) - not a solution. If x ≡ 1 (mod 3), then 2x ≡ 2 (mod 3) - not a solution. If x ≡ 2 (mod 3), then 2x ≡ 4 ≡ 1 (mod 3) - solution.

Thus, x must be of the form x 3k 2 for some integer k.

Step 3: Substituting and Simplifying

Substituting x 3k 2 back into the original equation:

2(3k 2) - 3y 763

6k 4 - 3y 763

3y 759 - 6k

y 253 - 2k

Step 4: Finding Bounds for k

For both x and y to be non-negative, we need:

For 3k 2 ≥ 0, k ≥ 0. For 253 - 2k ≥ 0, 253 ≥ 2k which simplifies to k ≤ 126.5, thus k ≤ 126.

The integer values of k range from 0 to 126 inclusive, giving us 127 possible values for k.

Conclusion

Therefore, the number of integral solutions to the equation 2x - 3y 763 is 127.

Further Considerations

It is important to note that there are infinitely many integral solutions to the equation 2x - 3y 763. These solutions are of the form x 1526 - 3n and y 2n - 763, where n is any integer. However, among these, only the positive integral solutions are considered here. For positive x and y, 382 ≤ n ≤ 508, which gives us 127 positive integral solutions.