Task: Cramer's rule
C++: #include <algorithm>
#include <iostream>
#include <vector>

class SubMatrix {
    const std::vector<std::vector<double>> *source;
    std::vector<double> replaceColumn;
    const SubMatrix *prev;
    size_t sz;
    int colIndex = -1;

public:
    SubMatrix(const std::vector<std::vector<double>> &src, const std::vector<double> &rc) : source(&src), replaceColumn(rc), prev(nullptr), colIndex(-1) {
        sz = replaceColumn.size();
    }

    SubMatrix(const SubMatrix &p) : source(nullptr), prev(&p), colIndex(-1) {
        sz = p.size() - 1;
    }

    SubMatrix(const SubMatrix &p, int deletedColumnIndex) : source(nullptr), prev(&p), colIndex(deletedColumnIndex) {
        sz = p.size() - 1;
    }

    int columnIndex() const {
        return colIndex;
    }
    void columnIndex(int index) {
        colIndex = index;
    }

    size_t size() const {
        return sz;
    }

    double index(int row, int col) const {
        if (source != nullptr) {
            if (col == colIndex) {
                return replaceColumn[row];
            } else {
                return (*source)[row][col];
            }
        } else {
            if (col < colIndex) {
                return prev->index(row + 1, col);
            } else {
                return prev->index(row + 1, col + 1);
            }
        }
    }

    double det() const {
        if (sz == 1) {
            return index(0, 0);
        }
        if (sz == 2) {
            return index(0, 0) * index(1, 1) - index(0, 1) * index(1, 0);
        }
        SubMatrix m(*this);
        double det = 0.0;
        int sign = 1;
        for (size_t c = 0; c < sz; ++c) {
            m.columnIndex(c);
            double d = m.det();
            det += index(0, c) * d * sign;
            sign = -sign;
        }
        return det;
    }
};

std::vector<double> solve(SubMatrix &matrix) {
    double det = matrix.det();
    if (det == 0.0) {
        throw std::runtime_error("The determinant is zero.");
    }

    std::vector<double> answer(matrix.size());
    for (int i = 0; i < matrix.size(); ++i) {
        matrix.columnIndex(i);
        answer[i] = matrix.det() / det;
    }
    return answer;
}

std::vector<double> solveCramer(const std::vector<std::vector<double>> &equations) {
    int size = equations.size();
    if (std::any_of(
        equations.cbegin(), equations.cend(),
        [size](const std::vector<double> &a) { return a.size() != size + 1; }
    )) {
        throw std::runtime_error("Each equation must have the expected size.");
    }

    std::vector<std::vector<double>> matrix(size);
    std::vector<double> column(size);
    for (int r = 0; r < size; ++r) {
        column[r] = equations[r][size];
        matrix[r].resize(size);
        for (int c = 0; c < size; ++c) {
            matrix[r][c] = equations[r][c];
        }
    }

    SubMatrix sm(matrix, column);
    return solve(sm);
}

template<typename T>
std::ostream &operator<<(std::ostream &os, const std::vector<T> &v) {
    auto it = v.cbegin();
    auto end = v.cend();

    os << '[';
    if (it != end) {
        os << *it++;
    }
    while (it != end) {
        os << ", " << *it++;
    }

    return os << ']';
}

int main() {
    std::vector<std::vector<double>> equations = {
        { 2, -1,  5,  1,  -3},
        { 3,  2,  2, -6, -32},
        { 1,  3,  3, -1, -47},
        { 5, -2, -3,  3,  49},
    };

    auto solution = solveCramer(equations);
    std::cout << solution << '\n';

    return 0;
}

Java: import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;

public class CramersRule {

    public static void main(String[] args) {
        Matrix mat = new Matrix(Arrays.asList(2d, -1d, 5d, 1d), 
                                Arrays.asList(3d, 2d, 2d, -6d), 
                                Arrays.asList(1d, 3d, 3d, -1d),
                                Arrays.asList(5d, -2d, -3d, 3d));
        List<Double> b = Arrays.asList(-3d, -32d, -47d, 49d);
        System.out.println("Solution = " + cramersRule(mat, b));
    }
    
    private static List<Double> cramersRule(Matrix matrix, List<Double> b) {
        double denominator = matrix.determinant();
        List<Double> result = new ArrayList<>();
        for ( int i = 0 ; i < b.size() ; i++ ) {
            result.add(matrix.replaceColumn(b, i).determinant() / denominator);
        }
        return result;
    }
        
    private static class Matrix {
        
        private List<List<Double>> matrix;
        
        @Override
        public String toString() {
            return matrix.toString();
        }
        
        @SafeVarargs
        public Matrix(List<Double> ... lists) {
            matrix = new ArrayList<>();
            for ( List<Double> list : lists) {
                matrix.add(list);
            }
        }
        
        public Matrix(List<List<Double>> mat) {
            matrix = mat;
        }
        
        public double determinant() {
            if ( matrix.size() == 1 ) {
                return get(0, 0);
            }
            if ( matrix.size() == 2 ) {
                return get(0, 0) * get(1, 1) - get(0, 1) * get(1, 0);
            }
            double sum = 0;
            double sign = 1;
            for ( int i = 0 ; i < matrix.size() ; i++ ) {
                sum += sign * get(0, i) * coFactor(0, i).determinant();
                sign *= -1;
            }
            return sum;
        }
        
        private Matrix coFactor(int row, int col) {
            List<List<Double>> mat = new ArrayList<>();
            for ( int i = 0 ; i < matrix.size() ; i++ ) {
                if ( i == row ) {
                    continue;
                }
                List<Double> list = new ArrayList<>();
                for ( int j = 0 ; j < matrix.size() ; j++ ) {
                    if ( j == col ) {
                        continue;
                    }
                    list.add(get(i, j));
                }
                mat.add(list);
            }
            return new Matrix(mat);
        }

        private Matrix replaceColumn(List<Double> b, int column) {
            List<List<Double>> mat = new ArrayList<>();
            for ( int row = 0 ; row < matrix.size() ; row++ ) {
                List<Double> list = new ArrayList<>();
                for ( int col = 0 ; col < matrix.size() ; col++ ) {
                    double value = get(row, col);
                    if ( col == column ) {
                        value = b.get(row);
                    }
                    list.add(value);
                }
                mat.add(list);
            }
            return new Matrix(mat);
        }

        private double get(int row, int col) {
            return matrix.get(row).get(col);
        }
        
    }

}