Solution for equation of a line with 2 variables ax + by = c works in one case but fails in other case

Question

I am trying to solve ax + by = n.

When I put n = 7, it solves the equation correctly as X = 2 and Y = 1. But when I put n = 1, it does not solve the equation. Even though, this equation has valid integer solution, X = 17, Y = -11. here is my full program.

#include <iostream>
using namespace std;

void PrintXY(int a, int b, int n)
{
    for (int i = 0; i * a <= n; i++) {

        if ((n - (i * a)) % b == 0) {
            cout << "x = " << i << ", y = "
                << (n - (i * a)) / b;
            return;
        }
    }

    cout << "No solution";
}

int main()
{
    int a = 2, b = 3, n = 1;
    PrintXY(a, b, n);
    return 0;
}

Output when n = 7:

x = 2, y = 1

Output when n = 1:

No solution

Reasoning. 2*(2) + 3*(1) - 7 = 4 + 3 - 7 = 0

2*(17) + 3*(-11) - 1 = 34 - 33 - 1 = 0

Both equations solve to give 0. But what is wrong in my program that is causing it to give "No Solution".

To pick at nits: To *solve* the linear equation, you need at least two equations. To **evaluate** the equation, you need to substitute values for the variables. — Thomas Matthews, Jan 20 '22 at 17:55
Note: infinity of solutions if `n` multiple of gcd of `a` and `b`, else no solution. One solution is directly provided by the extended Euclide algorithm. — Damien, Jan 20 '22 at 18:00
@ThomasMatthews Diophantine equation here. Can be solved, or not.. — Damien, Jan 20 '22 at 18:03
@ThomasMatthews, I am solving 1 varible, then reprzenting the other variable in terms on first variable. — Ganesh Kamath - 'Code Frenzy', Jan 20 '22 at 18:04
So you want to find integer solutions of the underdetermined equation? — Sebastian, Jan 20 '22 at 19:44
@Sebastian, solving this equation is one of the steps in software that I am unit testing. In the use cases, it is guaranteed that the solution has integer values. — Ganesh Kamath - 'Code Frenzy', Jan 21 '22 at 02:46

Mark B. · Answer 1 · 2022-01-20T18:15:14.680

2

The problem is with the termination condition:

i*a<=n

This(n>=a*i) need not be true, and is especially not true in case of the solution (ie X=17, Y=-11). Which seems reasonable - Without any bounds(limits) on the answer(either X or Y) , how would you find the solution to a linear equation(with an infinite possible range) in a closed for loop ?

edited Jan 20 '22 at 18:15

answered Jan 20 '22 at 17:57

Mark B.

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2
16

I am still only checking valid cases in my program. I will check infinite loop while writing invalid cases. – Ganesh Kamath - 'Code Frenzy' Jan 20 '22 at 18:03
I see, okay. Still, the termination condition needs to be less strict for you to hit upon your preferred integral solution. – Mark B. Jan 20 '22 at 18:05
Did you mean `Y=-11`? Did you miss a minus sign in your answer? as per the example a=2, b=3, n=1, `2 * 17 + 3 * (-11) = 1` – Wyck Jan 20 '22 at 18:11
I did, thanks for pointing it out. – Mark B. Jan 20 '22 at 18:15

Solution for equation of a line with 2 variables ax + by = c works in one case but fails in other case

1 Answers1