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)



Simultaneous Equations in Three Variables

Next we consider simultaneous equations in three unknowns or variables $x$ , $y$ and $z$. We pick another example from the same book Elementary Algebra2 by Hall & Knight.

$$6x + 2y - 5z = 13 \\ 3x + 3y - 2z = 13 \\ 7x + 5y - 3z = 26$$

We represent the above equations in the matrix form

$$\begin{bmatrix} 6 & 2 & -5 \\ 3 & 3 & -2 \\ 7 & 5 & -3 \\ \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \\ \end{bmatrix} = \begin{bmatrix} 13 \\ 13 \\ 26 \\ \end{bmatrix}$$

We solve it with NumPy's numpy.linalg.solve() function

					
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)



On executing the script, the solution is printed as a column matrix, representing the values for $x$ , $y$ and $z$ respectively.

					
[2. 3. 1.]



Notes

• 1) H. S. Hall & S. R. Knight, Elementary Algebra. London: Macmillan & Co., Ltd., 1896. Chapter XVII: Simultaneous Equations, p. 148, Ex. 2.
• 2) Ibid., p152