6.10. Systems of Linear Equations

Systems of linear equations can be solved with arrays and NumPy.

A system of linear equations is shown below:

\[ 8x + 3y -2z = 9 \]
\[ -4x + 7y + 5z = 15 \]
\[ 3x + 4y - 12z = 35 \]

NumPy’s np.linalg.solve() function can be used to solve this system of equations for the variables \(x\), \(y\) and \(z\).

The steps to solve the system of linear equations with np.linalg.solve() are below:

  • Create NumPy array A as a 3 by 3 array of the coefficients

  • Create a NumPy array b as the right-hand side of the equations

  • Solve for the values of \(x\), \(y\) and \(z\) using np.linalg.solve(A, b).

The resulting array has three entries. One entry for each variable.

import numpy as np

A = np.array([[8, 3, -2], [-4, 7, 5], [3, 4, -12]])
b = np.array([9, 15, 35])
x = np.linalg.solve(A, b)
x

array([-0.58226371,  3.22870478, -1.98599767])

We can plug the valuse of \(x\), \(y\) and \(z\) back into one of the equations to check the answer.

\(x\) is the first entry of the array, \(y\) is the second entry of the array, and \(z\) is the third entry of the array.

\(x\) = x[0]

\(y\) = x[1]

\(z\) = x[2]

When these values are plugged into the equation from above:

\[ 8x + 3y -2z = 9 \]

The answer should be 9.0.

8*x[0] + 3*x[1] - 2*x[2]

9.0