#### Simultaneous Linear Equation Solver

R
```"""
https://en.wikipedia.org/wiki/Augmented_matrix

This algorithm solves simultaneous linear equations of the form
λa + λb + λc + λd + ... = y as [λ, λ, λ, λ, ..., y]
Where λ & y 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 λ and y != 0
"""

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]]))
```