Hello friends, today I will talk about the consistency of equations. What is the consistency of the equations? How can I use determinant to check it out? Have a look!!

### The consistency of equations – what is it?

Suppose I have a set of 3 equations in two unknowns and . Now, these equations will be consistent if they have a common solution.

### How can I use determinant to check the consistency of equations?

In summary, the determinant of coefficients in 3 equations will be zero.

I just give you some examples. That will help!

Note: None of these examples are mine. I have chosen these from some book or books. I have also given the due reference at the end of the post.

So here is the first example.

##### Example 1

According to Stroud and Booth (2013)*, “Find the values of for which the following equations are consistent:

”

##### Solution

Here the given equations are

So I have 3 equations with 2 unknowns and . These equations will be consistent if the determinant of the coefficients will be zero.

Thus the first step will be to evaluate the determinant of the coefficients.

###### Step 1

The determinant of the coefficients is:

Now I will evaluate the determinant. Thus it will be

Here I can’t go further because I don’t know the values of .

Now I go to the next step.

###### Step 2

In this step, now I will equate the determinant to zero. After all, I have to get a set of consistent equations. Thus it will be

Now I will solve this equation for . This is a quadratic equation. So I will use the standard way to solve this.

Hence it will be like

Thus I can say either or .

For the value of will be

For the value of will be

So my conclusion is: for or , the equations will be consistent. This is the answer to this example.

Now I will show you another example.

##### Example 2

According to Stroud and Booth (2013)*, “Determine the values of for which the following equations have solutions:

”

##### Solution

Here the given equations are

So I have 3 equations with 2 unknowns and . Now if these equations have solutions, they will be consistent.

These equations will be consistent if the determinant of the coefficients will be zero.

Thus the first step will be to evaluate the determinant of the coefficients.

###### Step 1

The determinant of the coefficients is:

Now I will evaluate the determinant. Thus it will be

Here I can’t go further because I don’t know the values of .

Now I go to the next step.

###### Step 2

In this step, now I will equate the determinant to zero. After all, I have to get a set of consistent equations.

Thus it will be

Now I will solve this equation for . This is a quadratic equation. So I will use the standard way to solve this.

Hence it will be like

Thus I can say either or .

For the value of will be

For the value of will be

So my conclusion is: for or , the equations will be consistent.

This is the answer to this example.

Dear friends, this is the end of today’s example. Thank you very much for reading my post. Soon I will be back with a new post on a different topic!! Till then, bye, bye.

###### *Reference: K. A. Stroud and Dexter J. Booth (2013): Engineering mathematics, Industrial Press, Inc.; 7th Edition (March 8, 2013), Chapter: Determinants, Further problems 4, p. 486, Q. No.s 6 (Example 1), 7 (Example 2).

Ashish Pandit says

The definition of consistency is that the system should have at least one variable. Hence the definition makes all the HOMOGENOUS systems consistent.

Again, I think, consistency has got to do with Rank of Matrix and rank of augmented matrix, how did you use the value of Determinant then?

Ashish Pandit says

The definition of consistency is that the system should have at least one set of satisfying values of variables. Hence the definition makes all the HOMOGENOUS systems consistent.

Again, I think, consistency has got to do with Rank of Matrix and rank of augmented matrix, how did you use the value of Determinant then?