Swift Coding Test Course, Ax + By = C

Problem Approach

This problem is about determining whether a linear equation has a solution. There are various theories that can indicate whether a solution exists based on the values of A, B, and C, but we will approach it through a basic method commonly used in systems of equations.

Basic Theory

First, the equation Ax + By = C can be graphically interpreted in a two-dimensional coordinate system. If both A and B are 0, the equation does not hold, and if either A or B is 0, there is a single solution. Excluding these cases, typical scenarios can have different solutions.

The conditions for Ax + By = C to have a solution are as follows:

• If A = 0 and B = 0, there are infinitely many solutions if C is 0; otherwise, there is no solution.
• If A = 0, to have a solution (with respect to B), it must be a multiple of B.
• If B = 0, to have a solution (with respect to A), it must be a multiple of A.
• In all other cases, a solution exists.

Solution Algorithm

1. Check the existence of a solution using the values of A and B.
2. Output appropriately based on the existence of a solution.
3. If a solution exists, derive the solution using the slope and intercept of the line that includes the origin.

Swift Code

The following Swift code implements the algorithm described above.

import Foundation

func findSolution(A: Int, B: Int, C: Int) {
    if A == 0 && B == 0 {
        if C == 0 {
            print("A solution exists. (Infinitely many solutions)")
        } else {
            print("No solution exists.")
        }
    } else if A == 0 {
        // When B is not 0
        if C % B == 0 {
            let y = C / B
            print("A solution exists. (x is a free variable) -> x: any integer, y: \(y)")
        } else {
            print("No solution exists.")
        }
    } else if B == 0 {
        // When A is not 0
        if C % A == 0 {
            let x = C / A
            print("A solution exists. (y is a free variable) -> x: \(x), y: any integer")
        } else {
            print("No solution exists.")
        }
    } else {
        print("A solution exists. (Arbitrary solution - x: 0, y: \(C / B))")
    }
}

// Input values
let A = 2
let B = 3
let C = 6

findSolution(A: A, B: B, C: C)

Execution and Result

When the above Swift code is executed, it will output whether a solution exists for Ax + By = C. For example, if A = 2, B = 3, C = 6, the output will be as follows:

A solution exists. (Arbitrary solution - x: 0, y: 2)

Conclusion

In this tutorial, we explored the existence of solutions for linear equations in the form of Ax + By = C and the process of finding such solutions. By understanding the basics of algorithm problem-solving and writing code based on this, you can develop useful skills for coding tests or real programming situations.

Additionally, experimenting with various values of A, B, and C to analyze the patterns of solutions can be very helpful. In fact, providing various examples of both existing and non-existing solutions will greatly aid in understanding the problem.