Determining Consistency and Independence

Oct 13, 2025 by miodeya.com 133 views

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will focus on solving a system of two linear equations and determining whether the system is consistent and independent, consistent and dependent, or inconsistent.

The System of Equations

The given system of equations is:

{ \begin{array}{l} -4x + 7y = 28 \\ 4x - 7y = 32 \end{array} \}

Step 1: Write Down the System of Equations

The first step in solving the system of equations is to write down the equations in a clear and concise manner. In this case, we have two linear equations:

$-4x + 7y = 28$
$4x - 7y = 32$

Step 2: Add the Two Equations

To eliminate one of the variables, we can add the two equations together. This will result in a new equation that has only one variable.

{ \begin{array}{l} -4x + 7y = 28 \\ 4x - 7y = 32 \end{array} \}

Adding the two equations, we get:

$-4x + 7y + 4x - 7y = 28 + 32$

Simplifying the equation, we get:

$0 = 60$

Step 3: Analyze the Result

The resulting equation $0 = 60$ is a contradiction, as it is impossible for the sum of two numbers to be equal to 60 when both numbers are equal to 0. This means that the system of equations is inconsistent.

Conclusion

Based on the analysis, we can conclude that the system of equations is inconsistent. This means that there is no solution to the system, and the equations are not simultaneously true for any value of x and y.

Determining Consistency and Independence

A system of linear equations is said to be consistent if it has at least one solution. If the system has no solution, it is said to be inconsistent. A system of linear equations is said to be independent if the equations are not multiples of each other. If the equations are multiples of each other, the system is said to be dependent.

In this case, the system of equations is inconsistent, which means that it has no solution. Therefore, the system is not consistent and independent.

Final Answer

The final answer is:

The system of equations is inconsistent.
The system has no solution.
The equations are not multiples of each other.

Discussion

Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we focused on solving a system of two linear equations and determining whether the system is consistent and independent, consistent and dependent, or inconsistent. We used the method of addition to eliminate one of the variables and found that the system is inconsistent. This means that there is no solution to the system, and the equations are not simultaneously true for any value of x and y.

Example Use Cases

Solving a system of linear equations has many practical applications in various fields, including:

Physics: Solving a system of linear equations can help physicists determine the motion of objects under the influence of forces.
Engineering: Solving a system of linear equations can help engineers design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Solving a system of linear equations can help economists model and analyze economic systems, such as supply and demand.

Conclusion

In conclusion, solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we focused on solving a system of two linear equations and determining whether the system is consistent and independent, consistent and dependent, or inconsistent. We used the method of addition to eliminate one of the variables and found that the system is inconsistent. This means that there is no solution to the system, and the equations are not simultaneously true for any value of x and y.

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra: A Modern Introduction" by David Poole

Future Work

In the future, we can explore other methods for solving systems of linear equations, such as the method of substitution and the method of elimination. We can also apply these methods to more complex systems of linear equations and determine the consistency and independence of the systems.

Code

Here is some sample code in Python that can be used to solve a system of linear equations using the method of addition:

import numpy as np

# Define the coefficients of the equations
A = np.array([[-4, 7], [4, -7]])

# Define the constants of the equations
b = np.array([28, 32])

# Add the two equations together
result = np.add(A[0], A[1])

# Print the result
print(result)

# Check if the result is a contradiction
if np.all(result == 0):
    print("The system is inconsistent.")
else:
    print("The system is consistent.")

This code defines the coefficients of the equations and the constants of the equations, and then adds the two equations together using the np.add() function. It then checks if the result is a contradiction by checking if all the elements of the result are equal to 0. If the result is a contradiction, it prints "The system is inconsistent." Otherwise, it prints "The system is consistent."

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Introduction

In our previous article, we discussed how to solve a system of linear equations and determine whether the system is consistent and independent, consistent and dependent, or inconsistent. In this article, we will provide a Q&A guide to help you better understand the concepts and techniques involved in solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. Each equation is a linear combination of the variables, and the coefficients of the variables are constants.

Q: How do I determine whether a system of linear equations is consistent and independent, consistent and dependent, or inconsistent?

A: To determine whether a system of linear equations is consistent and independent, consistent and dependent, or inconsistent, you need to follow these steps:

Write down the system of equations in a clear and concise manner.
Add the two equations together to eliminate one of the variables.
Simplify the resulting equation and check if it is a contradiction.
If the resulting equation is a contradiction, the system is inconsistent.
If the resulting equation is not a contradiction, the system is consistent.
If the resulting equation is a multiple of one of the original equations, the system is dependent.
If the resulting equation is not a multiple of one of the original equations, the system is independent.

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

A: A consistent and independent system is a system that has a unique solution, and the equations are not multiples of each other. A consistent and dependent system is a system that has multiple solutions, and the equations are multiples of each other.

Q: How do I solve a system of linear equations using the method of substitution?

A: To solve a system of linear equations using the method of substitution, you need to follow these steps:

Write down the system of equations in a clear and concise manner.
Solve one of the equations for one of the variables.
Substitute the expression for the variable into the other equation.
Simplify the resulting equation and solve for the other variable.
Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: How do I solve a system of linear equations using the method of elimination?

A: To solve a system of linear equations using the method of elimination, you need to follow these steps:

Write down the system of equations in a clear and concise manner.
Add or subtract the two equations to eliminate one of the variables.
Simplify the resulting equation and solve for the other variable.
Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: What is the difference between the method of substitution and the method of elimination?

A: The method of substitution involves solving one of the equations for one of the variables and substituting the expression into the other equation. The method of elimination involves adding or subtracting the two equations to eliminate one of the variables.

Q: How do I determine whether a system of linear equations has a unique solution, multiple solutions, or no solution?

A: To determine whether a system of linear equations has a unique solution, multiple solutions, or no solution, you need to follow these steps:

Write down the system of equations in a clear and concise manner.
Add the two equations together to eliminate one of the variables.
Simplify the resulting equation and check if it is a contradiction.
If the resulting equation is a contradiction, the system has no solution.
If the resulting equation is not a contradiction, the system has a unique solution or multiple solutions.
If the resulting equation is a multiple of one of the original equations, the system has multiple solutions.
If the resulting equation is not a multiple of one of the original equations, the system has a unique solution.

Q: What are some common applications of systems of linear equations?

A: Systems of linear equations have many practical applications in various fields, including:

Physics: Solving systems of linear equations can help physicists determine the motion of objects under the influence of forces.
Engineering: Solving systems of linear equations can help engineers design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Solving systems of linear equations can help economists model and analyze economic systems, such as supply and demand.

Q: How do I use technology to solve systems of linear equations?

A: There are many software packages and online tools available that can help you solve systems of linear equations, including:

MATLAB
Mathematica
Python
Online calculators and graphing tools

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

Not following the order of operations
Not simplifying the equations correctly
Not checking for contradictions
Not using the correct method for solving the system

Conclusion

Solving systems of linear equations is an important skill in mathematics and has many practical applications in various fields. By following the steps outlined in this Q&A guide, you can learn how to solve systems of linear equations using the method of substitution and the method of elimination, and determine whether the system is consistent and independent, consistent and dependent, or inconsistent.

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra: A Modern Introduction" by David Poole

Future Work

In the future, we can explore other methods for solving systems of linear equations, such as the method of Gaussian elimination and the method of LU decomposition. We can also apply these methods to more complex systems of linear equations and determine the consistency and independence of the systems.

Code

Here is some sample code in Python that can be used to solve a system of linear equations using the method of substitution:

import numpy as np

# Define the coefficients of the equations
A = np.array([[2, 3], [4, 5]])

# Define the constants of the equations
b = np.array([6, 7])

# Solve one of the equations for one of the variables
x = np.linalg.solve(A, b)

# Print the solution
print(x)

# Substitute the expression for the variable into the other equation
y = np.linalg.solve(A, b)

# Print the solution
print(y)

This code defines the coefficients of the equations and the constants of the equations, and then solves the system of linear equations using the method of substitution. It then prints the solution to the system.