https:en.wikipedia.orgwikiAugmentedmatrix This algorithm solves simultaneous linear equations of the form a  b  c  d  ...   as , , , , ...,  Where    are individual coefficients, the no. of equations  no. of coefficients  1 Note in order to work there must exist 1 equation where all instances of  and  ! 0  simplify1, 2, 3, 4, 5, 6 1.0, 2.0, 3.0, 0.0, 0.75, 1.5  simplify5, 2, 5, 5, 1, 10 1.0, 0.4, 1.0, 0.0, 0.2, 1.0 Divide each row by magnitude of first term  creates 'unit' matrix Subtract to cancel term If first term is 0, it is already in form we want, so we preserve it Create next recursion iteration set  solvesimultaneous1, 2, 3,4, 5, 6 1.0, 2.0  solvesimultaneous0, 3, 1, 7,3, 2, 1, 11,5, 1, 2, 12 6.4, 1.2, 10.6  solvesimultaneous Traceback most recent call last: ... IndexError: solvesimultaneous requires n lists of length n1  solvesimultaneous1, 2, 3,1, 2 Traceback most recent call last: ... IndexError: solvesimultaneous requires n lists of length n1  solvesimultaneous1, 2, 3,a, 7, 8 Traceback most recent call last: ... ValueError: solvesimultaneous requires lists of integers  solvesimultaneous0, 2, 3,4, 0, 6 Traceback most recent call last: ... ValueError: solvesimultaneous requires at least 1 full equation
def simplify(current_set: list[list]) -> list[list]:
    """
    >>> simplify([[1, 2, 3], [4, 5, 6]])
    [[1.0, 2.0, 3.0], [0.0, 0.75, 1.5]]
    >>> simplify([[5, 2, 5], [5, 1, 10]])
    [[1.0, 0.4, 1.0], [0.0, 0.2, -1.0]]
    """
    # Divide each row by magnitude of first term --> creates 'unit' matrix
    duplicate_set = current_set.copy()
    for row_index, row in enumerate(duplicate_set):
        magnitude = row[0]
        for column_index, column in enumerate(row):
            if magnitude == 0:
                current_set[row_index][column_index] = column
                continue
            current_set[row_index][column_index] = column / magnitude
    # Subtract to cancel term
    first_row = current_set[0]
    final_set = [first_row]
    current_set = current_set[1::]
    for row in current_set:
        temp_row = []
        # If first term is 0, it is already in form we want, so we preserve it
        if row[0] == 0:
            final_set.append(row)
            continue
        for column_index in range(len(row)):
            temp_row.append(first_row[column_index] - row[column_index])
        final_set.append(temp_row)
    # Create next recursion iteration set
    if len(final_set[0]) != 3:
        current_first_row = final_set[0]
        current_first_column = []
        next_iteration = []
        for row in final_set[1::]:
            current_first_column.append(row[0])
            next_iteration.append(row[1::])
        resultant = simplify(next_iteration)
        for i in range(len(resultant)):
            resultant[i].insert(0, current_first_column[i])
        resultant.insert(0, current_first_row)
        final_set = resultant
    return final_set


def solve_simultaneous(equations: list[list]) -> list:
    """
    >>> solve_simultaneous([[1, 2, 3],[4, 5, 6]])
    [-1.0, 2.0]
    >>> solve_simultaneous([[0, -3, 1, 7],[3, 2, -1, 11],[5, 1, -2, 12]])
    [6.4, 1.2, 10.6]
    >>> solve_simultaneous([])
    Traceback (most recent call last):
        ...
    IndexError: solve_simultaneous() requires n lists of length n+1
    >>> solve_simultaneous([[1, 2, 3],[1, 2]])
    Traceback (most recent call last):
        ...
    IndexError: solve_simultaneous() requires n lists of length n+1
    >>> solve_simultaneous([[1, 2, 3],["a", 7, 8]])
    Traceback (most recent call last):
        ...
    ValueError: solve_simultaneous() requires lists of integers
    >>> solve_simultaneous([[0, 2, 3],[4, 0, 6]])
    Traceback (most recent call last):
        ...
    ValueError: solve_simultaneous() requires at least 1 full equation
    """
    if len(equations) == 0:
        raise IndexError("solve_simultaneous() requires n lists of length n+1")
    _length = len(equations) + 1
    if any(len(item) != _length for item in equations):
        raise IndexError("solve_simultaneous() requires n lists of length n+1")
    for row in equations:
        if any(not isinstance(column, (int, float)) for column in row):
            raise ValueError("solve_simultaneous() requires lists of integers")
    if len(equations) == 1:
        return [equations[0][-1] / equations[0][0]]
    data_set = equations.copy()
    if any(0 in row for row in data_set):
        temp_data = data_set.copy()
        full_row = []
        for row_index, row in enumerate(temp_data):
            if 0 not in row:
                full_row = data_set.pop(row_index)
                break
        if not full_row:
            raise ValueError("solve_simultaneous() requires at least 1 full equation")
        data_set.insert(0, full_row)
    useable_form = data_set.copy()
    simplified = simplify(useable_form)
    simplified = simplified[::-1]
    solutions: list = []
    for row in simplified:
        current_solution = row[-1]
        if not solutions:
            if row[-2] == 0:
                solutions.append(0)
                continue
            solutions.append(current_solution / row[-2])
            continue
        temp_row = row.copy()[: len(row) - 1 :]
        while temp_row[0] == 0:
            temp_row.pop(0)
        if len(temp_row) == 0:
            solutions.append(0)
            continue
        temp_row = temp_row[1::]
        temp_row = temp_row[::-1]
        for column_index, column in enumerate(temp_row):
            current_solution -= column * solutions[column_index]
        solutions.append(current_solution)
    final = []
    for item in solutions:
        final.append(float(round(item, 5)))
    return final[::-1]


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    eq = [
        [2, 1, 1, 1, 1, 4],
        [1, 2, 1, 1, 1, 5],
        [1, 1, 2, 1, 1, 6],
        [1, 1, 1, 2, 1, 7],
        [1, 1, 1, 1, 2, 8],
    ]
    print(solve_simultaneous(eq))
    print(solve_simultaneous([[4, 2]]))
