Consistent and Inconsistent Systems

Consistent Systems

A system is called consistent if there is at least one solution. Thus, the system from our initial example is a consistent system. It is possible that a system could have an infinite amount of solutions. Consider the following system:

If we solve the system algebraically we get the following:

elimination.png

As we see in this case we get 0 + 0 = 0, which is a true statement so the system is consistent. It turns out that both functions are the same line, thus the solution is anything that satisfies either one of the equations. When this happens we typically solve for one variable so we may say the solution is anything that satisfies the equation

You can choose any value for x and then find a corresponding y to make the system true.

Consider the following system:

This system would have the augmented matrix

matrix1.png

Use your calculator to get the matrix into reduced row echelon form.

Notice that the last row is all 0's. When that happens, we will have a consistent system that will have an infinite amount of solutions. In writing up our solution we will solve for our variables in terms of one of the others. With the way our matrix is written above it will be easiest to write our solutions in terms of z.

To do this take your reduced row echolon form matrix and write it in terms of the equations it represents then solve those equations in terms of one of the varaibles. So our solution would be

So our solution would be

Now for a given value of z we will calculate a particular x and y that make our system true.

Geometric Interpretation

In two dimensions when this happens it means we have the same line so they share all their points. In three dimensions it means that we have 3 planes intersecting at a line.

Inconsistent Systems

A system is inconsistent if it has no solutions.

Let's consider the following system:

Let's solve this system using elimination

Notice this time when we solve the system we get 0 = 4 which is an untrue statement. When this happens we call the system inconsistent. Graphically what is happening is there is no intersection between the two lines, they are parallel.

A system is inconsistent if in any row it has zeros in the coefficient side of the matrix and a number in the corresponding row of the column vector. The following augmented matrix would be inconsistent.

inconsistent_matrix1.png

Example

Consider the following system of equations:

Use your calculator to put this system into reduced row echelon form, what do you notice?

Here is a link that demonstrates how to show this system is inconsistent algebraically. For this course you do NOT need to be able to do this algebraically.

Geometric Interpretation

In 2 dimensions, when this happens we have two lines that are parallel (or do not intersect). In 3 dimensions, this means that our three planes do not intersect so at least two of them are parallel.

Examples

Here are two more examples of showing a systems is inconsistent algebraically.

Example 1

Example 2 

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