Which System Of Equations Has The Same Solution As The System Below?${ \begin{array}{l} 5x + 6y = -17 \ 3x + 3y = -12 \end{array} }$A. ${ \begin{array}{l} 5x + 6y = -17 \ -6x - 6y = 24 \end{array} }$B.

Mar 13, 2025 by ADMIN 204 views

**Which System of Equations Has the Same Solution as the Given System?**

Introduction

When dealing with systems of linear equations, it's often necessary to determine whether two systems have the same solution. This can be achieved by comparing the equations of the two systems and identifying any relationships between them. In this article, we will explore how to determine which system of equations has the same solution as the given system.

The Given System of Equations

The given system of equations is:

{ \begin{array}{l} 5x + 6y = -17 \\ 3x + 3y = -12 \end{array} \}

To begin, let's examine the equations of the given system. The first equation is $5x + 6y = -17$ , and the second equation is $3x + 3y = -12$ . We can see that the coefficients of $x$ and $y$ in the second equation are both $3$ , which is a multiple of the coefficients in the first equation.

Option A: System of Equations with Modified Second Equation

Option A presents a system of equations with a modified second equation:

{ \begin{array}{l} 5x + 6y = -17 \\ -6x - 6y = 24 \end{array} \}

Let's analyze the modified second equation. We can see that the coefficients of $x$ and $y$ are both $-6$ , which is the negative of the coefficients in the first equation. This suggests that the modified second equation is a multiple of the first equation.

Comparing the Systems of Equations

To determine whether the two systems have the same solution, we need to compare the equations of the two systems. We can start by multiplying the first equation of the given system by $-6$ to obtain:

$-30x - 36y = 102$

Now, let's compare this equation with the modified second equation in Option A:

$-6x - 6y = 24$

We can see that the coefficients of $x$ and $y$ in the two equations are the same, but the constant terms are different. However, we can rewrite the modified second equation as:

$-6x - 6y = 24$

$-6(5x + 6y) = -6(-17)$

$-30x - 36y = 102$

This shows that the modified second equation in Option A is equivalent to the first equation of the given system multiplied by $-6$ . Therefore, the two systems have the same solution.

Conclusion

In conclusion, the system of equations presented in Option A has the same solution as the given system. This is because the modified second equation in Option A is equivalent to the first equation of the given system multiplied by $-6$ . Therefore, the two systems have the same solution.

Discussion

The discussion category for this article is mathematics. The article explores how to determine which system of equations has the same solution as the given system. The article presents a system of equations and two options for a new system of equations. The article analyzes the equations of the two systems and compares them to determine whether the two systems have the same solution.

Key Takeaways

To determine whether two systems of equations have the same solution, we need to compare the equations of the two systems.
We can multiply the first equation of the given system by a constant to obtain a new equation.
If the new equation is equivalent to the second equation of the given system, then the two systems have the same solution.

Final Thoughts

Introduction

In our previous article, we explored how to determine which system of equations has the same solution as the given system. We analyzed the equations of the two systems and compared them to determine whether the two systems have the same solution. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the main difference between the two systems of equations?

A: The main difference between the two systems of equations is the second equation. In the given system, the second equation is $3x + 3y = -12$ , while in Option A, the second equation is $-6x - 6y = 24$ . However, as we showed in our previous article, the two equations are equivalent.

Q: How do we determine whether two systems of equations have the same solution?

A: To determine whether two systems of equations have the same solution, we need to compare the equations of the two systems. We can multiply the first equation of the given system by a constant to obtain a new equation. If the new equation is equivalent to the second equation of the given system, then the two systems have the same solution.

Q: What is the significance of multiplying the first equation by a constant?

A: Multiplying the first equation by a constant allows us to obtain a new equation that is equivalent to the second equation of the given system. This is a key step in determining whether the two systems have the same solution.

Q: Can we always multiply the first equation by a constant to obtain a new equation?

A: No, we cannot always multiply the first equation by a constant to obtain a new equation. The constant must be chosen such that the new equation is equivalent to the second equation of the given system.

Q: What if the two systems of equations have different solutions?

A: If the two systems of equations have different solutions, then they are not equivalent. In this case, we need to re-examine the equations of the two systems to determine the source of the difference.

Q: Can we use other methods to determine whether two systems of equations have the same solution?

A: Yes, we can use other methods to determine whether two systems of equations have the same solution. For example, we can use the method of substitution or the method of elimination to solve the systems of equations and compare the solutions.

Q: What is the importance of determining whether two systems of equations have the same solution?

A: Determining whether two systems of equations have the same solution is important in many areas of mathematics and science. For example, it is used in solving systems of linear equations, finding the intersection of two planes, and determining the stability of a system.

Conclusion

In conclusion, determining whether two systems of equations have the same solution is an important topic in mathematics and science. By comparing the equations of the two systems and using methods such as multiplication, substitution, and elimination, we can determine whether the two systems have the same solution. We hope that this article has provided a helpful overview of this topic and has answered some of the frequently asked questions related to it.

Key Takeaways

To determine whether two systems of equations have the same solution, we need to compare the equations of the two systems.
We can multiply the first equation of the given system by a constant to obtain a new equation.
If the new equation is equivalent to the second equation of the given system, then the two systems have the same solution.
We can use other methods such as substitution and elimination to determine whether two systems of equations have the same solution.
Determining whether two systems of equations have the same solution is important in many areas of mathematics and science.