NumPy: Solve a System of Linear Equations

We first consider a system of linear equations in two variables $x$ and $y$.

We pick an example from the classic Hall & Knight's text Elementary Algebra¹.

$$ 3x + 7y = 27 \\ 5x + 2y = 16 $$

This can be put in the matrix dot product form as

$$ \begin{bmatrix} 3 & 7 \\ 5 & 2 \\ \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} 27 \\ 16 \\ \end{bmatrix} $$

If $A$ represents the matrix of coefficients, $x$ the column vector of variables and $B$ the column vector of solutions, the above equation can be shortened to

$$ Ax = B $$

The above matrix product will be defined if and only if the number of columns in the coefficient matrix $A$ is equal to the number of rows in the variable matrix $x$.

From school, most of us are familiar with solving such set of linear equations using Cramer's Rule, which involves determinants.

Python's numerical library NumPy has a function numpy.linalg.solve() which solves a linear matrix equation, or system of linear scalar equation.

Here we find the solution to the above set of equations in Python using NumPy's numpy.linalg.solve() function.

				
					import numpy as np

					a = np.array([[3,7], [5,2]])
					b = np.array([27,16])
					x = np.linalg.solve(a, b)

					print(x)
				
			

If the above Python script is executed, we will get the solutions in the column matrix format as

				
					[2. 3.]
				
			

representing the $x$ and $y$ values respectively. So, $x = 2$ and $y = 3$.

Also you can use the numpy.allclose() function to check if the solution is correct. The following should return True.

				
					np.allclose(np.dot(a, x), b)
				
			

					
						import numpy as np

						a = np.array([[6,2,-5], [3,3,-2], [7,5,-3]])
						b = np.array([13,13,26])
						x = np.linalg.solve(a, b)

						print(x)
					
				

					
						[2. 3. 1.]