Systems of linear equations

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:

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]

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.

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.