In this tutorial, we will learn how to solve a system of linear equations in Python using Numpy. We will see two examples, first with a system of linear equations with two unknowns and. two variables. And the with a system of linear equations with three unknowns and three equations. We will use Numpy’s linalg.solve() function solve these equations.
The basic syntax of linalg.solve function is
linalg.solve(coeff_matrix, constant_vector)
System of Linear Equations
A system of linear equations consists of multiple linear equations with the same variables. Solving a system of linear equations requires finding the values of the unknown variables that satisfy all the equations simultaneously.
For example, here is a simple system of linear equations.
2x + y = 8 x + 3y = 14
Here we two equations with two unknown variables x and y. The aim of solving this system of linear equations is to find the values of x. and. y, for which the equations hold true.
For example, when x= 2 and y=4, both the first equation (2×2 + 4 = 8) and the. second equation (2 + 3*4 = 14) hold true. And we have found a solution for the system of equations.
We can write the system of equations in a matrix form
Ax = b
Where A is a 2×2 matrix with coefficients of the equation, x is the variables, and b are the constants.
A = np.array([[2, 1],
[1, 3]])
b = np.array([8,14])
Solving a System of Linear Equations with Numpy
We can use Numpy’s linalg.solve() function solve this system of linear equations in matrix form by providing the coefficient matrix A and b as arguments.
np.linalg.solve(A, b) [2,4]
And we get the same solution from Numpy’s linalg.solve as the one we found manually.
Solving a System of Linear Equations with 3 unknowns in Numpy
Numpy’s linalg.solve can be handy in a more complex system of linear equation. Here is an example of solving a system of linear equations with three variables.
3x + 2y - z = 1 2x - 2y + 4z = -2 -x + 0.5y -z = 0
As before, we can write this system of linear equations in matrix for Ax =b where A is
A = np.array([[3,2,-1],
[2,-2,4],
[-1,0.5,-1]])
Now we have the coefficients of the equations as a Numpy array.
A
array([[ 3. , 2. , -1. ],
[ 2. , -2. , 4. ],
[-1. , 0.5, -1. ]])
And the constant b is
b = np.array([1, -2,0]) b array([ 1, -2, 0])
With A and b in hour hand, we can use Numpy’s linalg.solve() function to solve for the values x,y,and z that will make the equations hold true.
np.linalg.solve(a, b) array([ 1., -2., -2.])
For our example, we get the solution as [1,-2,-2]. We can plugin the values and verify the system of linear equations.
In short, Numpy’s linalg.solve function is a convenient and efficient way to solve systems of linear equations in Python. It can be used to solve systems of equations with any number of variables, making it a useful tool for a wide range of applications.
