Determine If The Systems In Exercises 15 And 16 Are Consistent. Do Not Completely Solve The Systems.Exercise 15:$\[ \begin{aligned} x_1 - 6x_2 &= 5 \\ x_2 - 4x_3 + X_4 &= 0 \\ -x_1 + 6x_2 + X_3 + 5x_4 &= 3 \\ -x_2 + 5x_3 + 4x_4 &=

Introduction

In linear algebra, a system of linear equations is a collection of equations that are all linear, meaning they can be written in the form of a linear equation. These systems can be represented in matrix form, where the coefficients of the variables are the elements of the matrix. One of the fundamental concepts in linear algebra is the consistency of a system of linear equations. In this article, we will explore how to determine if the systems in Exercises 15 and 16 are consistent.

What is Consistency in Linear Systems?

Consistency in linear systems refers to the ability of the system to have a solution that satisfies all the equations simultaneously. A system is said to be consistent if it has at least one solution, while an inconsistent system has no solution. In other words, a consistent system is one where the equations are not contradictory, and there exists a set of values for the variables that satisfies all the equations.

Methods for Determining Consistency

There are several methods for determining the consistency of a system of linear equations. Some of the most common methods include:

• Gaussian Elimination: This method involves transforming the system of equations into row echelon form using elementary row operations. If the system is consistent, the row echelon form will have a unique solution. If the system is inconsistent, the row echelon form will have a contradiction, such as a row of zeros with a non-zero constant term.
• Matrix Inversion: This method involves finding the inverse of the coefficient matrix. If the system is consistent, the inverse will exist. If the system is inconsistent, the inverse will not exist.
• Rank and Nullity: This method involves finding the rank and nullity of the coefficient matrix. If the rank is equal to the number of variables, the system is consistent. If the rank is less than the number of variables, the system is inconsistent.

Exercise 15

The system of linear equations in Exercise 15 is given by:

$$\[ \begin{aligned} x_1 - 6x_2 &= 5 \\ x_2 - 4x_3 + x_4 &= 0 \\ -x_1 + 6x_2 + x_3 + 5x_4 &= 3 \\ -x_2 + 5x_3 + 4x_4 &= 0 \end{aligned}$$

To determine if this system is consistent, we can use the method of Gaussian elimination. We will transform the system into row echelon form using elementary row operations.

Step 1: Write the augmented matrix

The augmented matrix for the system is:

$$\[ \begin{aligned} \left[ \begin{array}{cccc|c} 1 & -6 & 0 & 0 & 5 \\ 0 & 1 & -4 & 1 & 0 \\ -1 & 6 & 1 & 5 & 3 \\ 0 & -1 & 5 & 4 & 0 \end{array} \right] \end{aligned}$$

Step 2: Perform elementary row operations

We will perform the following elementary row operations to transform the matrix into row echelon form:

• Swap rows 1 and 3
• Multiply row 1 by -1
• Add 6 times row 1 to row 2
• Add 1 times row 1 to row 4

The resulting matrix is:

$$\[ \begin{aligned} \left[ \begin{array}{cccc|c} -1 & 6 & 1 & 5 & 3 \\ 0 & 1 & -4 & 1 & 0 \\ 1 & -6 & 0 & 0 & 5 \\ 0 & -1 & 5 & 4 & 0 \end{array} \right] \end{aligned}$$

Step 3: Continue performing elementary row operations

We will continue performing elementary row operations to transform the matrix into row echelon form:

• Multiply row 2 by -1
• Add 6 times row 2 to row 1
• Add 1 times row 2 to row 4

The resulting matrix is:

$$\[ \begin{aligned} \left[ \begin{array}{cccc|c} -1 & 0 & 25 & 31 & 3 \\ 0 & 1 & -4 & 1 & 0 \\ 1 & -6 & 0 & 0 & 5 \\ 0 & 0 & 9 & 3 & 0 \end{array} \right] \end{aligned}$$

Step 4: Determine consistency

The final matrix is in row echelon form. We can see that the last row has a non-zero constant term, which indicates that the system is inconsistent.

Conclusion

In this article, we explored how to determine if the systems in Exercises 15 and 16 are consistent. We used the method of Gaussian elimination to transform the system into row echelon form and determined that the system in Exercise 15 is inconsistent. The method of Gaussian elimination is a powerful tool for determining the consistency of a system of linear equations. By transforming the system into row echelon form, we can easily determine if the system has a solution that satisfies all the equations simultaneously.

Exercise 16

The system of linear equations in Exercise 16 is given by:

$$\[ \begin{aligned} x_1 - 2x_2 + 3x_3 &= 0 \\ 2x_1 - 4x_2 + 6x_3 &= 0 \\ 3x_1 - 6x_2 + 9x_3 &= 0 \end{aligned}$$

To determine if this system is consistent, we can use the method of Gaussian elimination. We will transform the system into row echelon form using elementary row operations.

Step 1: Write the augmented matrix

The augmented matrix for the system is:

$$\[ \begin{aligned} \left[ \begin{array}{ccc|c} 1 & -2 & 3 & 0 \\ 2 & -4 & 6 & 0 \\ 3 & -6 & 9 & 0 \end{array} \right] \end{aligned}$$

Step 2: Perform elementary row operations

We will perform the following elementary row operations to transform the matrix into row echelon form:

• Multiply row 1 by -2
• Add 2 times row 1 to row 2
• Add 3 times row 1 to row 3

The resulting matrix is:

$$\[ \begin{aligned} \left[ \begin{array}{ccc|c} 1 & -2 & 3 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \end{aligned}$$

Step 3: Continue performing elementary row operations

We will continue performing elementary row operations to transform the matrix into row echelon form:

• Swap rows 1 and 2
• Multiply row 2 by -1

The resulting matrix is:

$$\[ \begin{aligned} \left[ \begin{array}{ccc|c} 0 & 0 & 0 & 0 \\ 1 & -2 & 3 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right] \end{aligned}$$

Step 4: Determine consistency

The final matrix is in row echelon form. We can see that the system has a row of zeros with a non-zero constant term, which indicates that the system is inconsistent.

Conclusion

Introduction

In our previous article, we explored how to determine if the systems in Exercises 15 and 16 are consistent. We used the method of Gaussian elimination to transform the system into row echelon form and determined that both systems are inconsistent. In this article, we will answer some frequently asked questions about determining consistency in linear systems.

Q: What is the difference between a consistent and inconsistent system?

A: A consistent system is one where the equations are not contradictory, and there exists a set of values for the variables that satisfies all the equations. An inconsistent system, on the other hand, has no solution, and the equations are contradictory.

Q: How do I determine if a system is consistent or inconsistent?

A: There are several methods for determining the consistency of a system of linear equations, including:

• Gaussian Elimination: This method involves transforming the system of equations into row echelon form using elementary row operations. If the system is consistent, the row echelon form will have a unique solution. If the system is inconsistent, the row echelon form will have a contradiction, such as a row of zeros with a non-zero constant term.
• Matrix Inversion: This method involves finding the inverse of the coefficient matrix. If the system is consistent, the inverse will exist. If the system is inconsistent, the inverse will not exist.
• Rank and Nullity: This method involves finding the rank and nullity of the coefficient matrix. If the rank is equal to the number of variables, the system is consistent. If the rank is less than the number of variables, the system is inconsistent.

Q: What are some common mistakes to avoid when determining consistency?

A: Some common mistakes to avoid when determining consistency include:

• Not transforming the system into row echelon form: Failing to transform the system into row echelon form can make it difficult to determine if the system is consistent or inconsistent.
• Not checking for contradictions: Failing to check for contradictions in the row echelon form can lead to incorrect conclusions about the consistency of the system.
• Not using the correct method: Using the wrong method for determining consistency can lead to incorrect conclusions.

Q: Can a system be both consistent and inconsistent?

A: No, a system cannot be both consistent and inconsistent. A system is either consistent or inconsistent, but not both.

Q: How do I know if a system is consistent or inconsistent without solving it?

A: You can determine if a system is consistent or inconsistent without solving it by using the methods mentioned above, such as Gaussian elimination, matrix inversion, or rank and nullity.

Q: Can a system have multiple solutions?

A: Yes, a system can have multiple solutions. However, if the system is inconsistent, it will have no solution.

Q: Can a system have no solution?

A: Yes, a system can have no solution. This is known as an inconsistent system.

Conclusion

In this article, we answered some frequently asked questions about determining consistency in linear systems. We discussed the difference between consistent and inconsistent systems, how to determine consistency, common mistakes to avoid, and more. By understanding these concepts, you can better determine if a system of linear equations is consistent or inconsistent.