Find Least Squares Solution of Inconsistent System
Consistent and Inconsistent Systems of Equations
All the systems of equations that we have seen in this section so far have had unique
solutions. These are referred to as Consistent Systems of Equations, meaning that
for a given system, there exists one solution set for the different variables in
the system or infinitely many sets of solution. In other words, as long as we can
find a solution for the system of equations, we refer to that system as being consistent
For a two variable system of equations to be consistent the lines formed by the
equations have to meet at some point or they have to be parallel.
For a three variable system of equations to be consistent, the equations formed
by the equations must meet two conditions:
- All three planes have to parallel
- Any two of the planes have to be parallel and the third must meet one of the planes
at some point and the other at another point.
Given that such systems exist, it is safe to conclude that Inconsistent systems
should exist as well, and they do. Inconsistent Systems of Equations are referred
to as such because for a given set of variables, there in no set of solutions for
the system of equations.
Inconsistent systems arise when the lines or planes formed from the systems of equations
don't meet at any point and are not parallel (all of them or only two and the
third meets one of the planes at some point.)
Two variable system of equations with Infinitely many solutions
The equations in a two variable system of equations are linear and hence can be
thought of as equations of two lines. When these two lines are parallel, then the
system has infinitely many solutions.
When two lines are parallel, their equations can usually be expressed as multiples
of each other and that's usually a quick way to spot systems with infinitely many
solutions.
For example, let's try to solve the system of equations below:
Using substitution method, we can solve for the variables as follows:
From equation (1)
substituting the above into equation(2)
In the above equation, we can see that we've lost all the variables from the equation.
This means that we can pick any value of x or y then substitute it
into any one of the two equations and then solve for the other variable.
For example if we pick x = 0, then if we substitute this into equation (1)
we would get y = 1. Any value we pick for x would give a different
value for y and thus there are infinitely many solutions for the system of
equations.
Two variable systems of equations with NO SOLUTION
There also exist two variable system of equations with no solution at all. This
happens when as we attempt to solve the system we end up an equation that makes
no sense mathematically.
For example, solve the system of equations below:
Using matrix method we can solve the above as follows:
Reducing the above to Row Echelon form can be done as follows:
Adding row 2 to row 1:
The equation formed from the second row of the matrix is given as
which means that:
But we know that the above is mathematically impossible. When we come across the
above, we say that the system of equations has NO SOLUTION. Thus we refer
to such systems as being inconsistent because they don't make any mathematical sense.
Three variable systems of equations with Infinite Solutions
When discussing the different methods of solving systems of equations, we only looked
at examples of systems with one unique solution set. These are known as Consistent
systems of equations but they are not the only ones. Three variable systems of equations
with infinitely many solution sets are also called consistent.
Since the equations in a three variable system of equations are linear, they can
also be thought of as equations of planes. The way these planes interact with each
other defines what kind of solution set they have and whether or not they have a
solution set. When these planes are parallel to each other, then the system of equations
that they form has infinitely many solutions.
Just as with two variable systems, three variable sytems have an infinte set of
solutions if when you solving for the variables you end up with an equation where
all the variables disappear.
For example; solve the system of equations below:
Solution:
Using matrix method:
In the last row of the above augmented matrix, we have ended up with all zeros on
both sides of the equations. This means that two of the planes formed by the equations
in the system of equations are parallel, and thus the system of equations is said
to have an infinite set of solutions. We solve for any of the set by assigning one
variable in the remaining two equations and then solving for the other two.
For example, if we take y =3
Then:
Then using the first row equation, we solve for x
Three variable systems with NO SOLUTION
Three variable systems of equations with no solution arise when the planed formed
by the equations in the system neither meet at point nor are they parallel. As a
result, when solving these systems, we end up with equations that make no mathematical
sense.
For example; solve the system of equations below
Solution:
Using Matrix method:
In the last row, we ended up with the equation 0 = 6 which we know can't
be true and so we conclude that the system of equations has no solution.
Find Least Squares Solution of Inconsistent System
Source: https://www.wyzant.com/resources/lessons/math/precalculus/systems_of_equations/consistent_and_inconsistent_systems/