The Condition for an Inconsistent System

Oct 13, 2025 by thesweetdesigns.com 108 views

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. The system is said to be consistent if it has at least one solution, and inconsistent if it has no solution. In this article, we will focus on finding the value of k for which the system of equations 3x - 7y = 1 and kx + 14y = 6 is inconsistent.

What is an Inconsistent System of Equations?

An inconsistent system of equations is a system that has no solution. This means that the equations are contradictory, and it is not possible to find a value of x and y that satisfies both equations simultaneously. In other words, the system is inconsistent if the equations are mutually exclusive, and there is no common solution that satisfies both equations.

The Condition for an Inconsistent System

For a system of two linear equations to be inconsistent, the two equations must be parallel. This means that the slopes of the two lines represented by the equations must be equal, but the y-intercepts must be different. In other words, the two lines must have the same slope, but different y-intercepts.

Finding the Value of k

To find the value of k for which the system of equations is inconsistent, we need to find the slope of the two lines represented by the equations. The slope of the first equation is 3/7, and the slope of the second equation is k/14. For the system to be inconsistent, the two slopes must be equal, but the y-intercepts must be different.

Solving for k

To solve for k, we can set the two slopes equal to each other and solve for k.

3/7 = k/14

To solve for k, we can cross-multiply and simplify the equation.

3(14) = 7k

42 = 7k

Now, we can divide both sides of the equation by 7 to solve for k.

k = 42/7

k = 6

Conclusion

In conclusion, the value of k for which the system of equations 3x - 7y = 1 and kx + 14y = 6 is inconsistent is 6. This means that when k = 6, the two equations are parallel, and there is no common solution that satisfies both equations simultaneously.

Final Answer

The final answer is C. 6.

Additional Information

It's worth noting that the value of k can also be expressed as a fraction. To express k as a fraction, we can simplify the equation k = 42/7.

k = 6

k = 6/1

k = 6

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra and Its Applications" by David C. Lay

Glossary

Inconsistent System: A system of linear equations that has no solution.
Consistent System: A system of linear equations that has at least one solution.
Slope: A measure of the steepness of a line.
Y-intercept: The point at which a line intersects the y-axis.

Online Resources

For online resources on linear algebra and its applications, we recommend the following websites:

[1] Khan Academy: Linear Algebra
[2] MIT OpenCourseWare: Linear Algebra
[3] Wolfram MathWorld: Linear Algebra

Q: What is an inconsistent system of equations?

A: An inconsistent system of equations is a system that has no solution. This means that the equations are contradictory, and it is not possible to find a value of x and y that satisfies both equations simultaneously.

Q: How can I determine if a system of equations is inconsistent?

A: To determine if a system of equations is inconsistent, you can check if the two equations are parallel. This means that the slopes of the two lines represented by the equations must be equal, but the y-intercepts must be different.

Q: What is the condition for a system of equations to be inconsistent?

A: For a system of two linear equations to be inconsistent, the two equations must be parallel. This means that the slopes of the two lines represented by the equations must be equal, but the y-intercepts must be different.

Q: How do I find the value of k for which the system of equations is inconsistent?

A: To find the value of k for which the system of equations is inconsistent, you can set the two slopes equal to each other and solve for k. This will give you the value of k for which the system of equations is inconsistent.

Q: What is the final answer for the value of k?

A: The final answer for the value of k is 6.

Q: Can I express the value of k as a fraction?

A: Yes, you can express the value of k as a fraction. To do this, you can simplify the equation k = 42/7.

Q: What are some additional resources for learning about linear algebra and its applications?

A: Some additional resources for learning about linear algebra and its applications include:

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon
[3] "Linear Algebra and Its Applications" by David C. Lay

Q: Where can I find online resources for learning about linear algebra and its applications?

A: Some online resources for learning about linear algebra and its applications include:

[1] Khan Academy: Linear Algebra
[2] MIT OpenCourseWare: Linear Algebra
[3] Wolfram MathWorld: Linear Algebra

Q: What are some related topics to linear algebra and its applications?

A: Some related topics to linear algebra and its applications include:

Linear Equations: Equations that involve a linear relationship between two or more variables.
Systems of Linear Equations: A set of two or more linear equations that involve two or more variables.
Linear Algebra: A branch of mathematics that deals with the study of linear equations and linear transformations.

Q: What are some key terms related to inconsistent systems of equations?

A: Some key terms related to inconsistent systems of equations include:

Inconsistent System: A system of linear equations that has no solution.
Consistent System: A system of linear equations that has at least one solution.
Slope: A measure of the steepness of a line.
Y-intercept: The point at which a line intersects the y-axis.

Q: How can I further my knowledge of linear algebra and its applications?

A: To further your knowledge of linear algebra and its applications, you can:

Read books on linear algebra and its applications, such as "Linear Algebra and Its Applications" by Gilbert Strang.
Take online courses on linear algebra and its applications, such as those offered by Khan Academy and MIT OpenCourseWare.
Practice solving problems and exercises on linear algebra and its applications.
Join online communities and forums to discuss linear algebra and its applications with others.