






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





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


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 lowerCamelCase			:					Tuple         =	[
     [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]]))














from __future__ import annotations





def       __lowerCAmelCase      (						__snake_case					, __snake_case					, __snake_case       ):
     __lowerCAmelCase								=					list(range(len(__snake_case       )       )       )
     __lowerCAmelCase								=					[v / w for v, w in zip(__snake_case					, __snake_case       )]
     index.sort(key=lambda __snake_case       : ratio[i]					, reverse=__snake_case       )

     __lowerCAmelCase								=					0
     __lowerCAmelCase								=					[0] * len(__snake_case       )
     for i in index:
          if weight[i] <= capacity:
               __lowerCAmelCase								=					1
               max_value += value[i]
               capacity -= weight[i]
          else:
               __lowerCAmelCase								=					capacity / weight[i]
               max_value += value[i] * capacity / weight[i]
               break

     return max_value, fractions


if __name__ == "__main__":
 import doctest

 doctest.testmod()






