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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:

  1. All three planes have to parallel
  2. 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.