#### Introduction

- A system of linear equations is of the form:
- Each row is an individual equation.
- Variables
**x**_{1} to **x**_{n} are shared amongst the equations. - The size of this system is denoted
**m** x **n**.

#### Example

- Two cars are in a race
- The first car is slow:
- It can travel at 10 meters per second
- But it has a 1 minute head start

- The second car is fast:
- It can travel at 25 meters per second
- But it has no head start

- The cars are not accelerating
- Their distance as an equation of time forms a straight line
- and is therefore a linear equation

- We can represent them as a system of linear equations:
**t** is time in seconds**d** is the distance travelled

- By solving these equations we can find the point where the fast car overtakes the slow one
- The cars meet when the distance each has travelled is equal The cars meet after 40 seconds
- We can solve for
**d** by putting 40 into one of the original equations: The cars meet at 1000 meters - Finally, we can re-arrange these equations to make them look more like the generic one found in the introduction:

#### Code (Python)

```
import numpy as np
A = np.array([[1, -10], [1, -25]])
B = np.array([600, 0])
solution = np.linalg.solve(A, B)
print(solution) # prints [1000, 40]
```

#### Dependent vs Independent equations

- Equations are independent when each of them gives us new information about a system.
- For example: is dependent because one is simply a multiple of another, and tells us nothing new.

#### Consistent vs Inconsistent systems

- A solution for a linear equation is a set of values which make the equation true.
- For example:
- One solution would be:
- But it can also be solved with:

- Similarly, a solution for a system of linear equations is a set of values which make all the equations true:
- For example:
- One solution is:
- However, any set of values of the form: is also a solution.

- Some systems will only have a single solution.
- Such as the one in the example.

- Some systems will not have any solutions:
- For example:
- There are no values we can give for
**x** and **y** that makes all 3 equations true.

- The set of all solutions for a system of equations is called the solution set.
- If the set is empty, then the system is inconsistent.
- If the set has at least one solution, then the system is consistent.