


def 			lowerCAmelCase (		lowerCAmelCase_ )-> list[list]:
				lowerCAmelCase_ :      Optional[int]    				=				current_set.copy()
				for row_index, row in enumerate(lowerCAmelCase_ ):
								lowerCAmelCase_ :      Dict    				=				row[0]
								for column_index, column in enumerate(lowerCAmelCase_ ):
												if magnitude == 0:
																lowerCAmelCase_ :      List[Any]    				=				column
																continue
												lowerCAmelCase_ :      Union[str, Any]    				=				column / magnitude
    # Subtract to cancel term
				lowerCAmelCase_ :      Tuple    				=				current_set[0]
				lowerCAmelCase_ :      str    				=				[first_row]
				lowerCAmelCase_ :      Union[str, Any]    				=				current_set[1::]
				for row in current_set:
								lowerCAmelCase_ :      Tuple    				=				[]
								# If first term is 0, it is already in form we want, so we preserve it
								if row[0] == 0:
												final_set.append(lowerCAmelCase_ )
												continue
								for column_index in range(len(lowerCAmelCase_ ) ):
												temp_row.append(first_row[column_index] - row[column_index] )
								final_set.append(lowerCAmelCase_ )
				# Create next recursion iteration set
				if len(final_set[0] ) != 3:
								lowerCAmelCase_ :      Union[str, Any]    				=				final_set[0]
								lowerCAmelCase_ :      Optional[int]    				=				[]
								lowerCAmelCase_ :      int    				=				[]
								for row in final_set[1::]:
												current_first_column.append(row[0] )
												next_iteration.append(row[1::] )
								lowerCAmelCase_ :      List[str]    				=				simplify(lowerCAmelCase_ )
								for i in range(len(lowerCAmelCase_ ) ):
												resultant[i].insert(0 ,   current_first_column[i] )
								resultant.insert(0 ,   lowerCAmelCase_ )
								lowerCAmelCase_ :      Optional[Any]    				=				resultant
				return final_set



def 			lowerCAmelCase (		lowerCAmelCase_ )-> list:
				if len(lowerCAmelCase_ ) == 0:
								raise IndexError('''solve_simultaneous() requires n lists of length n+1''' )
				lowerCAmelCase_ :      Any    				=				len(lowerCAmelCase_ ) + 1
				if any(len(lowerCAmelCase_ ) != _length for item in equations ):
								raise IndexError('''solve_simultaneous() requires n lists of length n+1''' )
				for row in equations:
								if any(not isinstance(lowerCAmelCase_ ,   (int, float) ) for column in row ):
												raise ValueError('''solve_simultaneous() requires lists of integers''' )
				if len(lowerCAmelCase_ ) == 1:
								return [equations[0][-1] / equations[0][0]]
				lowerCAmelCase_ :      Optional[Any]    				=				equations.copy()
				if any(0 in row for row in data_set ):
								lowerCAmelCase_ :      int    				=				data_set.copy()
								lowerCAmelCase_ :      str    				=				[]
								for row_index, row in enumerate(lowerCAmelCase_ ):
												if 0 not in row:
																lowerCAmelCase_ :      Any    				=				data_set.pop(lowerCAmelCase_ )
																break
								if not full_row:
												raise ValueError('''solve_simultaneous() requires at least 1 full equation''' )
								data_set.insert(0 ,   lowerCAmelCase_ )
				lowerCAmelCase_ :      Any    				=				data_set.copy()
				lowerCAmelCase_ :      str    				=				simplify(lowerCAmelCase_ )
				lowerCAmelCase_ :      Any    				=				simplified[::-1]
				lowerCAmelCase_ :      list    				=				[]
				for row in simplified:
								lowerCAmelCase_ :      Dict    				=				row[-1]
								if not solutions:
												if row[-2] == 0:
																solutions.append(0 )
																continue
												solutions.append(current_solution / row[-2] )
												continue
								lowerCAmelCase_ :      str    				=				row.copy()[: len(lowerCAmelCase_ ) - 1 :]
								while temp_row[0] == 0:
												temp_row.pop(0 )
								if len(lowerCAmelCase_ ) == 0:
												solutions.append(0 )
												continue
								lowerCAmelCase_ :      List[str]    				=				temp_row[1::]
								lowerCAmelCase_ :      int    				=				temp_row[::-1]
								for column_index, column in enumerate(lowerCAmelCase_ ):
												current_solution -= column * solutions[column_index]
								solutions.append(lowerCAmelCase_ )
				lowerCAmelCase_ :      Optional[int]    				=				[]
				for item in solutions:
								final.append(float(round(lowerCAmelCase_ ,   5 ) ) )
				return final[::-1]


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						_UpperCAmelCase			:  List[Any]     =[
						    [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]]))



_UpperCAmelCase			:  Dict     =[
    (1000, """M"""),
    (900, """CM"""),
    (500, """D"""),
    (400, """CD"""),
    (100, """C"""),
    (90, """XC"""),
    (50, """L"""),
    (40, """XL"""),
    (10, """X"""),
    (9, """IX"""),
    (5, """V"""),
    (4, """IV"""),
    (1, """I"""),
]



def 			lowerCAmelCase (		lowerCAmelCase_ )-> int:
				lowerCAmelCase_ :      Any    				=				{'''I''': 1, '''V''': 5, '''X''': 10, '''L''': 50, '''C''': 100, '''D''': 500, '''M''': 1_000}
				lowerCAmelCase_ :      Optional[int]    				=				0
				lowerCAmelCase_ :      List[str]    				=				0
				while place < len(lowerCAmelCase_ ):
								if (place + 1 < len(lowerCAmelCase_ )) and (vals[roman[place]] < vals[roman[place + 1]]):
												total += vals[roman[place + 1]] - vals[roman[place]]
												place += 2
								else:
												total += vals[roman[place]]
												place += 1
				return total



def 			lowerCAmelCase (		lowerCAmelCase_ )-> str:
				lowerCAmelCase_ :      List[Any]    				=				[]
				for arabic, roman in ROMAN:
								((lowerCAmelCase_)				,  (lowerCAmelCase_)) :      Optional[int]    				=				divmod(lowerCAmelCase_ ,   lowerCAmelCase_ )
								result.append(roman * factor )
								if number == 0:
												break
				return "".join(lowerCAmelCase_ )


if __name__ == "__main__":
						import doctest

						doctest.testmod()