Inconsistent systems of equations

Hey today we are going to look at how to recognize system of equations with no or infinitely many solution after we have done some consistent systems of equations!
There are four circumstances:

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.

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.