





def    __UpperCamelCase      (    a,     a) ->tuple[float, float]:

					# Check if the input is valid
					if not len(a) == len(a) == 3:
										raise ValueError("Please enter a valid equation.")
					if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
										raise ValueError("Both a & b of two equations can't be zero.")

					# Extract the coefficients
					lowerCamelCase__    ,		lowerCamelCase__    ,		lowerCamelCase__       =       equationa
					lowerCamelCase__    ,		lowerCamelCase__    ,		lowerCamelCase__       =       equationa

					# Calculate the determinants of the matrices
					lowerCamelCase__       =       aa * ba - aa * ba
					lowerCamelCase__       =       ca * ba - ca * ba
					lowerCamelCase__       =       aa * ca - aa * ca

					# Check if the system of linear equations has a solution (using Cramer's rule)
					if determinant == 0:
										if determinant_x == determinant_y == 0:
															raise ValueError("Infinite solutions. (Consistent system)")
										else:
															raise ValueError("No solution. (Inconsistent system)")
					else:
										if determinant_x == determinant_y == 0:
															# Trivial solution (Inconsistent system)
															return (0.0, 0.0)
										else:
															lowerCamelCase__       =       determinant_x / determinant
															lowerCamelCase__       =       determinant_y / determinant
															# Non-Trivial Solution (Consistent system)
															return (x, y)








from __future__ import annotations

import math
import random
from collections.abc import Collection
from typing import overload

class SCREAMING_SNAKE_CASE_ :





					"""simple docstring"""


					def __init__( self	,  _lowerCAmelCase = None	):
										if components is None:
															lowerCamelCase__       =       []
										lowerCamelCase__       =       list(_lowerCAmelCase	)





					def __len__( self	):
										return len(self.__components	)





					def __str__( self	):
										return "(" + ",".join(map(_lowerCAmelCase	,  self.__components	)	) + ")"





					def __add__( self	,  _lowerCAmelCase	):
										lowerCamelCase__       =       len(self	)
										if size == len(_lowerCAmelCase	):
															lowerCamelCase__       =       [self.__components[i] + other.component(_lowerCAmelCase	) for i in range(_lowerCAmelCase	)]
															return Vector(_lowerCAmelCase	)
										else:
															raise Exception("must have the same size"	)





					def __sub__( self	,  _lowerCAmelCase	):
										lowerCamelCase__       =       len(self	)
										if size == len(_lowerCAmelCase	):
															lowerCamelCase__       =       [self.__components[i] - other.component(_lowerCAmelCase	) for i in range(_lowerCAmelCase	)]
															return Vector(_lowerCAmelCase	)
										else:  # error case
															raise Exception("must have the same size"	)





					@overload
					def __mul__( self	,  _lowerCAmelCase	):
										...





					@overload
					def __mul__( self	,  _lowerCAmelCase	):
										...





					def __mul__( self	,  _lowerCAmelCase	):
										if isinstance(_lowerCAmelCase	,  (float, int)	):
															lowerCamelCase__       =       [c * other for c in self.__components]
															return Vector(_lowerCAmelCase	)
										elif isinstance(_lowerCAmelCase	,  _lowerCAmelCase	) and len(self	) == len(_lowerCAmelCase	):
															lowerCamelCase__       =       len(self	)
															lowerCamelCase__       =       [self.__components[i] * other.component(_lowerCAmelCase	) for i in range(_lowerCAmelCase	)]
															return sum(_lowerCAmelCase	)
										else:  # error case
															raise Exception("invalid operand!"	)





					def 					__magic_name__ ( self	):
										return Vector(self.__components	)





					def 					__magic_name__ ( self	,  _lowerCAmelCase	):
										if isinstance(_lowerCAmelCase	,  _lowerCAmelCase	) and -len(self.__components	) <= i < len(self.__components	):
															return self.__components[i]
										else:
															raise Exception("index out of range"	)





					def 					__magic_name__ ( self	,  _lowerCAmelCase	,  _lowerCAmelCase	):
										assert -len(self.__components	) <= pos < len(self.__components	)
										lowerCamelCase__       =       value





					def 					__magic_name__ ( self	):
										if len(self.__components	) == 0:
															raise Exception("Vector is empty"	)
										lowerCamelCase__       =       [c**2 for c in self.__components]
										return math.sqrt(sum(_lowerCAmelCase	)	)






					def 					__magic_name__ ( self	,  _lowerCAmelCase	,  _lowerCAmelCase = False	):
										lowerCamelCase__       =       self * other
										lowerCamelCase__       =       self.euclidean_length() * other.euclidean_length()
										if deg:
															return math.degrees(math.acos(num / den	)	)
										else:
															return math.acos(num / den	)





def    __UpperCamelCase      (    a) ->Vector:
					assert isinstance(a,     a)
					return Vector([0] * dimension)





def    __UpperCamelCase      (    a,     a) ->Vector:
					assert isinstance(a,     a) and (isinstance(a,     a))
					lowerCamelCase__       =       [0] * dimension
					lowerCamelCase__       =       1
					return Vector(a)





def    __UpperCamelCase      (    a,     a,     a) ->Vector:
					assert (
					    isinstance(a,     a)
					    and isinstance(a,     a)
					    and (isinstance(a,     (int, float)))
					)
					return x * scalar + y





def    __UpperCamelCase      (    a,     a,     a) ->Vector:
					random.seed(a)
					lowerCamelCase__       =       [random.randint(a,     a) for _ in range(a)]
					return Vector(a)

class SCREAMING_SNAKE_CASE_ :





					"""simple docstring"""


					def __init__( self	,  _lowerCAmelCase	,  _lowerCAmelCase	,  _lowerCAmelCase	):
										lowerCamelCase__       =       matrix
										lowerCamelCase__       =       w
										lowerCamelCase__       =       h





					def __str__( self	):
										lowerCamelCase__       =       ""
										for i in range(self.__height	):
															ans += "|"
															for j in range(self.__width	):
																				if j < self.__width - 1:
																									ans += str(self.__matrix[i][j]	) + ","
																				else:
																									ans += str(self.__matrix[i][j]	) + "|\n"
										return ans





					def __add__( self	,  _lowerCAmelCase	):
										if self.__width == other.width() and self.__height == other.height():
															lowerCamelCase__       =       []
															for i in range(self.__height	):
																				lowerCamelCase__       =       [
																				    self.__matrix[i][j] + other.component(_lowerCAmelCase	,  _lowerCAmelCase	)
																				    for j in range(self.__width	)
																				]
																				matrix.append(_lowerCAmelCase	)
															return Matrix(_lowerCAmelCase	,  self.__width	,  self.__height	)
										else:
															raise Exception("matrix must have the same dimension!"	)





					def __sub__( self	,  _lowerCAmelCase	):
										if self.__width == other.width() and self.__height == other.height():
															lowerCamelCase__       =       []
															for i in range(self.__height	):
																				lowerCamelCase__       =       [
																				    self.__matrix[i][j] - other.component(_lowerCAmelCase	,  _lowerCAmelCase	)
																				    for j in range(self.__width	)
																				]
																				matrix.append(_lowerCAmelCase	)
															return Matrix(_lowerCAmelCase	,  self.__width	,  self.__height	)
										else:
															raise Exception("matrices must have the same dimension!"	)





					@overload
					def __mul__( self	,  _lowerCAmelCase	):
										...





					@overload
					def __mul__( self	,  _lowerCAmelCase	):
										...





					def __mul__( self	,  _lowerCAmelCase	):
										if isinstance(_lowerCAmelCase	,  _lowerCAmelCase	):  # matrix-vector
															if len(_lowerCAmelCase	) == self.__width:
																				lowerCamelCase__       =       zero_vector(self.__height	)
																				for i in range(self.__height	):
																									lowerCamelCase__       =       [
																									    self.__matrix[i][j] * other.component(_lowerCAmelCase	)
																									    for j in range(self.__width	)
																									]
																									ans.change_component(_lowerCAmelCase	,  sum(_lowerCAmelCase	)	)
																				return ans
															else:
																				raise Exception(
																				    "vector must have the same size as the "
																				    "number of columns of the matrix!"	)
										elif isinstance(_lowerCAmelCase	,  (int, float)	):  # matrix-scalar
															lowerCamelCase__       =       [
															    [self.__matrix[i][j] * other for j in range(self.__width	)]
															    for i in range(self.__height	)
															]
															return Matrix(_lowerCAmelCase	,  self.__width	,  self.__height	)
										return None





					def 					__magic_name__ ( self	):
										return self.__height





					def 					__magic_name__ ( self	):
										return self.__width





					def 					__magic_name__ ( self	,  _lowerCAmelCase	,  _lowerCAmelCase	):
										if 0 <= x < self.__height and 0 <= y < self.__width:
															return self.__matrix[x][y]
										else:
															raise Exception("change_component: indices out of bounds"	)





					def 					__magic_name__ ( self	,  _lowerCAmelCase	,  _lowerCAmelCase	,  _lowerCAmelCase	):
										if 0 <= x < self.__height and 0 <= y < self.__width:
															lowerCamelCase__       =       value
										else:
															raise Exception("change_component: indices out of bounds"	)





					def 					__magic_name__ ( self	,  _lowerCAmelCase	,  _lowerCAmelCase	):
										if self.__height != self.__width:
															raise Exception("Matrix is not square"	)
										lowerCamelCase__       =       self.__matrix[:x] + self.__matrix[x + 1 :]
										for i in range(len(_lowerCAmelCase	)	):
															lowerCamelCase__       =       minor[i][:y] + minor[i][y + 1 :]
										return Matrix(_lowerCAmelCase	,  self.__width - 1	,  self.__height - 1	).determinant()





					def 					__magic_name__ ( self	,  _lowerCAmelCase	,  _lowerCAmelCase	):
										if self.__height != self.__width:
															raise Exception("Matrix is not square"	)
										if 0 <= x < self.__height and 0 <= y < self.__width:
															return (-1) ** (x + y) * self.minor(_lowerCAmelCase	,  _lowerCAmelCase	)
										else:
															raise Exception("Indices out of bounds"	)






					def 					__magic_name__ ( self	):
										if self.__height != self.__width:
															raise Exception("Matrix is not square"	)
										if self.__height < 1:
															raise Exception("Matrix has no element"	)
										elif self.__height == 1:
															return self.__matrix[0][0]
										elif self.__height == 2:
															return (
															    self.__matrix[0][0] * self.__matrix[1][1]
															    - self.__matrix[0][1] * self.__matrix[1][0]
															)
										else:
															lowerCamelCase__       =       [
															    self.__matrix[0][y] * self.cofactor(0	,  _lowerCAmelCase	) for y in range(self.__width	)
															]
															return sum(_lowerCAmelCase	)





def    __UpperCamelCase      (    a) ->Matrix:
					lowerCamelCase__       =       [[0] * n for _ in range(a)]
					return Matrix(a,     a,     a)







def    __UpperCamelCase      (    a,     a,     a,     a) ->Matrix:
					random.seed(a)
					lowerCamelCase__       =       [
					    [random.randint(a,     a) for _ in range(a)] for _ in range(a)
					]
					return Matrix(a,     a,     a)

