
def  lowerCAmelCase__			(  SCREAMING_SNAKE_CASE_       ,							SCREAMING_SNAKE_CASE_     )				->		tuple[float, float]:

			# Check if the input is valid
			if not len(SCREAMING_SNAKE_CASE_     ) == len(SCREAMING_SNAKE_CASE_     ) == 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__      :      str         =		equationa
			lowerCAmelCase__      ,				lowerCAmelCase__      ,				lowerCAmelCase__      :      Dict         =		equationa

			# Calculate the determinants of the matrices
			lowerCAmelCase__      :      Any         =		aa * ba - aa * ba
			lowerCAmelCase__      :      Optional[int]         =		ca * ba - ca * ba
			lowerCAmelCase__      :      Optional[int]         =		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__      :      Optional[Any]         =		determinant_x / determinant
									lowerCAmelCase__      :      Any         =		determinant_y / determinant
									# Non-Trivial Solution (Consistent system)
									return (x, y)

from __future__ import annotations

lowerCamelCase__          =		list[list[int]]

# assigning initial values to the grid
lowerCamelCase__          =		[
    [3, 0, 6, 5, 0, 8, 4, 0, 0],
    [5, 2, 0, 0, 0, 0, 0, 0, 0],
    [0, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]

# a grid with no solution
lowerCamelCase__          =		[
    [5, 0, 6, 5, 0, 8, 4, 0, 3],
    [5, 2, 0, 0, 0, 0, 0, 0, 2],
    [1, 8, 7, 0, 0, 0, 0, 3, 1],
    [0, 0, 3, 0, 1, 0, 0, 8, 0],
    [9, 0, 0, 8, 6, 3, 0, 0, 5],
    [0, 5, 0, 0, 9, 0, 6, 0, 0],
    [1, 3, 0, 0, 0, 0, 2, 5, 0],
    [0, 0, 0, 0, 0, 0, 0, 7, 4],
    [0, 0, 5, 2, 0, 6, 3, 0, 0],
]
def  lowerCAmelCase__			(  SCREAMING_SNAKE_CASE_       ,							SCREAMING_SNAKE_CASE_       ,							SCREAMING_SNAKE_CASE_       ,							SCREAMING_SNAKE_CASE_     )				->		bool:
			for i in range(9     ):
						if grid[row][i] == n or grid[i][column] == n:
									return False

			for i in range(3     ):
						for j in range(3     ):
									if grid[(row - row % 3) + i][(column - column % 3) + j] == n:
												return False

			return True
def  lowerCAmelCase__			(  SCREAMING_SNAKE_CASE_     )				->		tuple[int, int] | None:
			for i in range(9     ):
						for j in range(9     ):
									if grid[i][j] == 0:
												return i, j
			return None
def  lowerCAmelCase__			(  SCREAMING_SNAKE_CASE_     )				->		Matrix | None:
			if location := find_empty_location(SCREAMING_SNAKE_CASE_     ):
						lowerCAmelCase__      ,				lowerCAmelCase__      :      Any         =		location
			else:
						# If the location is ``None``, then the grid is solved.
						return grid

			for digit in range(1       ,							10     ):
						if is_safe(SCREAMING_SNAKE_CASE_       ,							SCREAMING_SNAKE_CASE_       ,							SCREAMING_SNAKE_CASE_       ,							SCREAMING_SNAKE_CASE_     ):
									lowerCAmelCase__      :      Optional[Any]         =		digit

									if sudoku(SCREAMING_SNAKE_CASE_     ) is not None:
												return grid

									lowerCAmelCase__      :      List[Any]         =		0

			return None
def  lowerCAmelCase__			(  SCREAMING_SNAKE_CASE_     )				->		None:
			for row in grid:
						for cell in row:
									print(SCREAMING_SNAKE_CASE_       ,							end=' '     )
						print()


if __name__ == "__main__":
					# make a copy of grid so that you can compare with the unmodified grid
					for example_grid in (initial_grid, no_solution):
										print("""\nExample grid:\n""" + """=""" * 20)
										print_solution(example_grid)
										print("""\nExample grid solution:""")
										lowerCamelCase__          =		sudoku(example_grid)
										if solution is not None:
															print_solution(solution)
										else:
															print("""Cannot find a solution.""")