Conditions for a System to Have One Solution

Oct 13, 2025 by windailys.com 79 views

Introduction

When dealing with systems of equations, it's essential to understand the conditions that lead to a unique solution. In this article, we will explore the concept of a system of equations having one 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 goal is to find the values of the variables that satisfy all the equations in the system. In this case, we have the equation $4x - y = 5$ , and we need to find the other equation that, when combined with this one, will result in a system with one solution.

Conditions for a System to Have One Solution

For a system of equations to have one solution, the two equations must be linearly independent. This means that the equations must not be parallel to each other, and the lines must intersect at exactly one point. In other words, the slopes of the two lines must be different.

Understanding the Given Equation

The given equation is $4x - y = 5$ . To understand the possibilities for the other equation, we need to rewrite this equation in slope-intercept form, which is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

y = 4x - 5

Analyzing the Options

Now, let's analyze the options for the other equation:

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

y = -4x + 5

This equation has a slope of $-4$ , which is different from the slope of the given equation, $4$ . Therefore, this option is a possible candidate for the other equation in the system.

Option B: $y = 4x - 5$

y = 4x - 5

This equation has the same slope as the given equation, $4$ . Since the slopes are the same, the lines will be parallel, and the system will not have a unique solution. Therefore, this option is not a possible candidate for the other equation in the system.

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

2y = 8x - 10

To rewrite this equation in slope-intercept form, we need to divide both sides by $2$ .

y = 4x - 5

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

-2y = -8x - 10

To rewrite this equation in slope-intercept form, we need to divide both sides by $-2$ .

y = 4x + 5

This equation has a slope of $4$ , which is the same as the slope of the given equation. However, the y-intercepts are different, which means that the lines will intersect at exactly one point. Therefore, this option is a possible candidate for the other equation in the system.

Conclusion

In conclusion, for a system of equations to have one solution, the two equations must be linearly independent. The given equation is $4x - y = 5$ , and we need to find the other equation that, when combined with this one, will result in a system with one solution. The possible options for the other equation are $y = -4x + 5$ and $y = 4x + 5$ . These options meet the conditions for a system to have one solution, and the lines will intersect at exactly one point.

Frequently Asked Questions

What is a system of equations?
What are the conditions for a system to have one solution?
How do I determine if two equations are linearly independent?
What is the slope-intercept form of an equation?

References

Introduction

In our previous article, we explored the concept of a system of equations having one 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 provide a Q&A section to help clarify any doubts and provide further understanding of the topic.

Q&A

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 goal is to find the values of the variables that satisfy all the equations in the system.

Q: What are the conditions for a system to have one solution?

A: For a system of equations to have one solution, the two equations must be linearly independent. This means that the equations must not be parallel to each other, and the lines must intersect at exactly one point.

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

A: To determine if two equations are linearly independent, you can rewrite them in slope-intercept form and compare their slopes. If the slopes are different, the equations are linearly independent.

Q: What is the slope-intercept form of an equation?

A: The slope-intercept form of an equation is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Q: Can a system of equations have more than one solution?

A: Yes, a system of equations can have more than one solution. This occurs when the equations are linearly dependent, meaning that the lines are parallel and intersect at multiple points.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution. This occurs when the equations are inconsistent, meaning that the lines do not intersect at any point.

Q: How do I solve a system of equations?

A: To solve a system of equations, you can use various methods such as substitution, elimination, or graphing. The method you choose will depend on the type of equations and the number of variables.

Q: What is the difference between a system of equations and a system of inequalities?

A: A system of equations consists of two or more equations that contain multiple variables, whereas a system of inequalities consists of two or more inequalities that contain multiple variables.

Q: Can a system of inequalities have a solution?

A: Yes, a system of inequalities can have a solution. This occurs when the inequalities are consistent, meaning that the solution set is not empty.

Q: Can a system of inequalities have no solution?

A: Yes, a system of inequalities can have no solution. This occurs when the inequalities are inconsistent, meaning that the solution set is empty.

Conclusion

In conclusion, a system of equations with one solution requires the two equations to be linearly independent. The given equation is $4x - y = 5$ , and we need to find the other equation that, when combined with this one, will result in a system with one solution. The possible options for the other equation are $y = -4x + 5$ and $y = 4x + 5$ . These options meet the conditions for a system to have one solution, and the lines will intersect at exactly one point.

Frequently Asked Questions

What is a system of equations?
What are the conditions for a system to have one solution?
How do I determine if two equations are linearly independent?
What is the slope-intercept form of an equation?
Can a system of equations have more than one solution?
Can a system of equations have no solution?