A System Of Equations Has 1 Solution. If 4 X − Y = 5 4x - Y = 5 4 X − Y = 5 Is One Of The Equations, Which Could Be The Other Equation?A. Y = − 4 X + 5 Y = -4x + 5 Y = − 4 X + 5 B. Y = 4 X − 5 Y = 4x - 5 Y = 4 X − 5 C. 2 Y = 8 X − 10 2y = 8x - 10 2 Y = 8 X − 10 D. − 2 Y = − 8 X − 10 -2y = -8x - 10 − 2 Y = − 8 X − 10

Introduction

When dealing with systems of equations, it's essential to understand the relationship between the equations and their solutions. In this article, we will explore the concept of a system of equations having 1 solution and how it relates to the given equation $4x - y = 5$. We will examine the possible options for the other equation in the system and discuss the conditions that must be met for the system to have a unique solution.

What is a System of Equations?

A system of equations is a set of two or more equations that contain multiple variables. The system is said to be consistent if it has at least one solution, and it is said to be inconsistent if it has no solution. In this article, we are interested in systems of equations that have a unique solution, meaning that there is only one possible solution that satisfies both equations.

The Given Equation

The given equation is $4x - y = 5$. This equation represents a line in the coordinate plane, and it has a slope of 4 and a y-intercept of -5. To find the other equation in the system, we need to consider the conditions that must be met for the system to have a unique solution.

Conditions for a Unique Solution

For a system of equations to have a unique solution, the two equations must be linearly independent. This means that the equations must not be parallel or identical. In other words, the equations must have different slopes or y-intercepts.

Analyzing the Options

Let's analyze the options for the other equation in the system:

Option A: $y = -4x + 5$

This equation represents a line with a slope of -4 and a y-intercept of 5. Since the slope of this equation is the negative reciprocal of the slope of the given equation, the two equations are parallel. Therefore, this option does not meet the condition for a unique solution.

Option B: $y = 4x - 5$

This equation represents a line with the same slope as the given equation, but with a different y-intercept. Since the slopes are the same, the two equations are identical. Therefore, this option does not meet the condition for a unique solution.

Option C: $2y = 8x - 10$

This equation represents a line with a slope of 4 and a y-intercept of -5. Since the slope and y-intercept of this equation are the same as the given equation, the two equations are identical. Therefore, this option does not meet the condition for a unique solution.

Option D: $-2y = -8x - 10$

This equation represents a line with a slope of 4 and a y-intercept of -5. Since the slope and y-intercept of this equation are the same as the given equation, the two equations are identical. Therefore, this option does not meet the condition for a unique solution.

Conclusion

Based on the analysis of the options, none of the given equations meet the condition for a unique solution. However, we can modify the equations to meet the condition. For example, we can multiply the given equation by 2 to get $8x - 2y = 10$. This equation is linearly independent of the given equation, and the system has a unique solution.

Final Answer

The final answer is not among the given options. However, we can conclude that the other equation in the system must be linearly independent of the given equation. This means that the other equation must have a different slope or y-intercept than the given equation.

Discussion

The discussion of this problem highlights the importance of understanding the relationship between the equations and their solutions. By analyzing the options and considering the conditions for a unique solution, we can determine the correct answer. This problem requires a deep understanding of linear algebra and the properties of systems of equations.

Additional Resources

For more information on systems of equations and linear algebra, please refer to the following resources:

• Linear Algebra
• Systems of Equations
• Linear Independence

Conclusion

In conclusion, a system of equations has 1 solution if the two equations are linearly independent. The given equation $4x - y = 5$ represents a line in the coordinate plane, and the other equation in the system must be linearly independent of this equation. By analyzing the options and considering the conditions for a unique solution, we can determine the correct answer. This problem requires a deep understanding of linear algebra and the properties of systems of equations.

Introduction

In our previous article, we explored the concept of a system of equations having 1 solution and how it relates to the given equation $4x - y = 5$. We analyzed the possible options for the other equation in the system and discussed the conditions that must be met for the system to have a unique solution. In this article, we will answer some frequently asked questions (FAQs) related to this topic.

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that contain multiple variables. The system is said to be consistent if it has at least one solution, and it is said to be inconsistent if it has no solution.

Q: What is the condition for a system of equations to have a unique solution?

A: For a system of equations to have a unique solution, the two equations must be linearly independent. This means that the equations must not be parallel or identical. In other words, the equations must have different slopes or y-intercepts.

Q: What is the relationship between the slopes of the two equations in a system of equations?

A: The slopes of the two equations in a system of equations must be different for the system to have a unique solution. If the slopes are the same, the equations are parallel, and the system has no solution. If the slopes are different, the equations are linearly independent, and the system has a unique solution.

Q: How can I determine if two equations are linearly independent?

A: To determine if two equations are linearly independent, you can check if the slopes of the equations are different. If the slopes are different, the equations are linearly independent. If the slopes are the same, the equations are parallel, and the system has no solution.

Q: What is the significance of the y-intercept in a system of equations?

A: The y-intercept of an equation represents the point where the equation intersects the y-axis. In a system of equations, the y-intercepts of the two equations must be different for the system to have a unique solution.

Q: Can a system of equations have a unique solution if the equations are identical?

A: No, a system of equations cannot have a unique solution if the equations are identical. If the equations are identical, the system has no solution.

Q: Can a system of equations have a unique solution if the equations are parallel?

A: No, a system of equations cannot have a unique solution if the equations are parallel. If the equations are parallel, the system has no solution.

Q: How can I modify an equation to make it linearly independent of another equation?

A: To modify an equation to make it linearly independent of another equation, you can multiply the equation by a constant. For example, if you have the equation $y = 2x + 3$, you can multiply it by 2 to get $2y = 4x + 6$. This new equation is linearly independent of the original equation.

Q: What is the final answer to the problem?

A: The final answer to the problem is not among the given options. However, we can conclude that the other equation in the system must be linearly independent of the given equation. This means that the other equation must have a different slope or y-intercept than the given equation.

Conclusion

In conclusion, a system of equations has 1 solution if the two equations are linearly independent. The given equation $4x - y = 5$ represents a line in the coordinate plane, and the other equation in the system must be linearly independent of this equation. By analyzing the options and considering the conditions for a unique solution, we can determine the correct answer. This problem requires a deep understanding of linear algebra and the properties of systems of equations.

Additional Resources

For more information on systems of equations and linear algebra, please refer to the following resources:

• Linear Algebra
• Systems of Equations
• Linear Independence

Final Thoughts

In this article, we answered some frequently asked questions related to the concept of a system of equations having 1 solution. We discussed the conditions that must be met for a system of equations to have a unique solution and provided examples to illustrate the concepts. We hope that this article has been helpful in clarifying the concepts and providing a deeper understanding of linear algebra and the properties of systems of equations.