C#

Thank goodness for high school algebra!

using System.Diagnostics;
using Common;

namespace Day13;

static class Program
{
    static void Main()
    {
        var start = Stopwatch.GetTimestamp();

        var sampleInput = Input.ParseInput("sample.txt");
        var programInput = Input.ParseInput("input.txt");

        Console.WriteLine($"Part 1 sample: {Part1(sampleInput)}");
        Console.WriteLine($"Part 1 input: {Part1(programInput)}");

        Console.WriteLine($"Part 2 sample: {Part2(sampleInput)}");
        Console.WriteLine($"Part 2 input: {Part2(programInput)}");

        Console.WriteLine($"That took about {Stopwatch.GetElapsedTime(start)}");
    }

    static object Part1(Input i) => i.Machines
        .Select(m => CalculateCoins(m, 100))
        .Where(c => c > 0)
        .Sum();

    static object Part2(Input i) => i.Machines
        .Select(m => m with { Prize = new XYValues(
            m.Prize.X + 10000000000000,
            m.Prize.Y + 10000000000000) })
        .Select(m => CalculateCoins(m, long.MaxValue))
        .Where(c => c > 0)
        .Sum();

    static long CalculateCoins(Machine m, long limit)
    {
        var bBottom = (m.A.X * m.B.Y) - (m.A.Y * m.B.X);
        if (bBottom == 0) return -1;

        var bTop = (m.Prize.Y * m.A.X) - (m.Prize.X * m.A.Y);
        var b = bTop / bBottom;
        if ((b <= 0) || (b > limit)) return -1;
        
        var a = (m.Prize.X - (b * m.B.X)) / m.A.X;
        if ((a <= 0) || (a > limit)) return -1;

        var calcPrizeX = (a * m.A.X) + (b * m.B.X);
        var calcPrizeY = (a * m.A.Y) + (b * m.B.Y);
        if ((m.Prize.X != calcPrizeX) || (m.Prize.Y != calcPrizeY)) return -1;

        return (3 * a) + b;
    }
}

public record struct Machine(XYValues A, XYValues B, XYValues Prize);
public record struct XYValues(long X, long Y);

public class Input
{
    private Input()
    {
    }

    public List<Machine> Machines { get; init; }

    public static Input ParseInput(string file)
    {
        var machines = new List<Machine>();

        var lines = File.ReadAllLines(file);
        for(int l=0; l<lines.Length; l+=4)
        {
            machines.Add(new Machine(
                ParseXYValues(lines[l + 0]),
                ParseXYValues(lines[l + 1]),
                ParseXYValues(lines[l + 2])));
        }

        return new Input()
        {
            Machines = machines,
        };
    }

    private static readonly char[] XYDelimiters = ['X', 'Y', '=', '+', ',', ' '];

    private static XYValues ParseXYValues(string s)
    {
        var parts = s
            .Substring(s.IndexOf(':') + 1)
            .SplitAndTrim(XYDelimiters)
            .Select(long.Parse)
            .ToArray();
        
        return new XYValues(parts[0], parts[1]);
    }
}