**A system of linear equations (SLE)** is a set of equations where each equation is a linear combination of variables. The goal is to find a solution that satisfies all the equations simultaneously.   
## Example
Let's look at the example of a system of linear equations:

We have 3 unknown variables `x`, `y` and `z` and have 3 equations that include all of these variables.   
## Solving the system
How can we solve the system? Firstly, let's rewrite it in a matrix form:

Expressing the system of linear equations in matrix form provides us with a straightforward approach for solving the system utilizing the inverse matrix:

To use this approach we have to be sure that matrix A can be inversed:   
**The square matrix A can be inversed if and only if its determinant is nonzero**.    
## Example 1

import numpy as np

# Define the coefficients matrix A
A = np.array([[2, 3, -1],
              [4, -1, 2],
              [1, 2, -3]])

# Define the constants vector н
y = np.array([10, 4, -1])

# Check the determinant of matrix A
print(f'Determinant is {round(np.linalg.det(A), 3)}')

# Calculating inversed matrix
A_inv = np.linalg.inv(A)
# Finding solution using inversed matrix
x = np.dot(A_inv, y)
    
# Print the solution
print(f'Solution: {np.round(x, 3)}')


We have found the solution using the inverted matrix. 
## Example 2

import numpy as np

# Define the coefficients matrix A
A = np.array([[1, 2, 3],
              [3, 1, 4],
              [4, 3, 7]])

# Define the constants vector b
b = np.array([10, 20, 30])

# Check the determinant of matrix A
det_A = np.linalg.det(A)

# Print the determinant value
print(f'Determinant of A: {det_A}')

A_inv = np.linalg.inv(A)

The code above produces an error - the matrix is **singular (has zero determinant)** so we can't solve the system of equations.    
The explanation for this is quite simple: the matrix rows are **linearly dependent** (the third row is the sum of the first two). As a result, **the third equation doesn't provide any additional information**, and we are left with a system of 3 variables but only 2 unique equations. As a result such a system either have no solutions or there are lots of solutions.

To study many applied disciplines, it is necessary to have basic knowledge of higher mathematics and linear algebra. Mathematics is often used in data analysis, system modeling, building machine learning models, etc. Let's look at some of the most important topics and learn how to use the mathematical apparatus to solve real problems.

Let's start with some basic definitions and concepts we'll use later. Consider the idea of a function, a numerical sequence, and its sum, and also understand what a coordinate system's basis is.

The simplest and most commonly used type of relationship is the linear relationship. Linear algebra is a branch of higher mathematics entirely devoted to linear functions and linear spaces. Let's look at some of the most important topics in linear algebra: vectors, matrices, solving linear equations, and solving the spectral problem for matrices.

Mathematical analysis is a discipline that allows you to analyze functions according to various criteria. Consider how to check numerical sequences for convergence, find the maximum/minimum values of functions, solve nonlinear equations, and use integrals to solve applied problems.

Mathematics for Data Analysis and Modeling

System of Linear Equations

Example

Solving the system

Example 1

Example 2