Is ( 3 , − 4 (3,-4 ( 3 , − 4 ] A Solution To This System Of Equations? 5 X + 2 Y = 7 8 X + 5 Y = 4 \begin{array}{l} 5x + 2y = 7 \\ 8x + 5y = 4 \end{array} 5 X + 2 Y = 7 8 X + 5 Y = 4 ​ A. YesB. No

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Introduction

When dealing with systems of linear equations, it is essential to determine whether a given point satisfies both equations. In this case, we are presented with a system of two linear equations and asked to verify if the point $(3,-4)$ is a solution to the system. To do this, we will substitute the coordinates of the point into both equations and check if the resulting statements are true.

Understanding the System of Equations

The given system of equations consists of two linear equations in the form of:

$\begin{array}{l} 5x + 2y = 7 \\ 8x + 5y = 4 \end{array}$

To determine if the point $(3,-4)$ is a solution, we need to substitute $x=3$ and $y=-4$ into both equations and check if the resulting statements are true.

Substituting the Point into the First Equation

Let's start by substituting $x=3$ and $y=-4$ into the first equation:

$5x + 2y = 7$

Substituting the values, we get:

$5(3) + 2(-4) = 7$

Expanding and simplifying the equation, we get:

$15 - 8 = 7$

Simplifying further, we get:

$7 = 7$

This statement is true, which means that the point $(3,-4)$ satisfies the first equation.

Substituting the Point into the Second Equation

Now, let's substitute $x=3$ and $y=-4$ into the second equation:

$8x + 5y = 4$

Substituting the values, we get:

$8(3) + 5(-4) = 4$

Expanding and simplifying the equation, we get:

$24 - 20 = 4$

Simplifying further, we get:

$4 = 4$

This statement is also true, which means that the point $(3,-4)$ satisfies the second equation.

Conclusion

Since the point $(3,-4)$ satisfies both equations, we can conclude that it is a solution to the system of equations.

Final Answer

The final answer is: A. Yes

Discussion

In this discussion, we have demonstrated how to determine if a given point is a solution to a system of linear equations. By substituting the coordinates of the point into both equations and checking if the resulting statements are true, we can verify if the point is a solution to the system. This is an essential skill in mathematics, particularly in algebra and geometry.

Importance of Checking Solutions

Checking solutions to a system of equations is crucial in mathematics, as it helps us to:

• Verify the accuracy of our work
• Identify any errors or inconsistencies in the system
• Understand the behavior of the system and its solutions
• Make informed decisions based on the results

Real-World Applications

The concept of checking solutions to a system of equations has numerous real-world applications, including:

• Physics and engineering: to model and analyze complex systems
• Economics: to understand the behavior of markets and economies
• Computer science: to develop algorithms and solve problems
• Data analysis: to interpret and visualize data

Conclusion

In conclusion, checking solutions to a system of equations is an essential skill in mathematics, and it has numerous real-world applications. By understanding how to verify solutions, we can gain a deeper understanding of the system and its behavior, and make informed decisions based on the results.

Final Thoughts

The concept of checking solutions to a system of equations is a fundamental aspect of mathematics, and it requires a deep understanding of algebra and geometry. By mastering this skill, we can develop a strong foundation in mathematics and apply it to real-world problems and applications.

References

• [1] "Algebra and Geometry" by Michael Artin
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• Khan Academy: Systems of Linear Equations
• MIT OpenCourseWare: Linear Algebra
• Wolfram Alpha: Systems of Linear Equations
  

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Introduction

In our previous article, we explored whether the point $(3,-4)$ is a solution to the system of equations:

$\begin{array}{l} 5x + 2y = 7 \\ 8x + 5y = 4 \end{array}$

We concluded that the point $(3,-4)$ is indeed a solution to the system. In this Q&A article, we will address some common questions and concerns related to this topic.

Q: What is a solution to a system of equations?

A: A solution to a system of equations is a point that satisfies all the equations in the system. In other words, it is a point that makes all the equations true.

Q: How do I determine if a point is a solution to a system of equations?

A: To determine if a point is a solution to a system of equations, you need to substitute the coordinates of the point into each equation and check if the resulting statement is true.

Q: What if the point does not satisfy one of the equations?

A: If the point does not satisfy one of the equations, then it is not a solution to the system. You can try to find the solution by using other methods, such as graphing or substitution.

Q: Can a system of equations have multiple solutions?

A: Yes, a system of equations can have multiple solutions. This occurs when the system has infinitely many solutions, which means that there are an infinite number of points that satisfy all the equations.

Q: How do I find the solution to a system of equations with multiple solutions?

A: To find the solution to a system of equations with multiple solutions, you need to use a method such as graphing or substitution to find the general solution. The general solution is an equation that describes all the possible solutions to the system.

Q: What if the system of equations has no solution?

A: If the system of equations has no solution, then it means that there is no point that satisfies all the equations. This can occur when the equations are inconsistent or contradictory.

Q: How do I determine if a system of equations has no solution?

A: To determine if a system of equations has no solution, you need to check if the equations are inconsistent or contradictory. You can do this by using methods such as substitution or graphing.

Q: Can a system of equations have a unique solution?

A: Yes, a system of equations can have a unique solution. This occurs when the system has a single point that satisfies all the equations.

Q: How do I find the solution to a system of equations with a unique solution?

A: To find the solution to a system of equations with a unique solution, you need to use a method such as substitution or elimination to find the solution.

Q: What if I am given a system of equations and I am not sure if it has a solution?

A: If you are given a system of equations and you are not sure if it has a solution, you can try to use methods such as graphing or substitution to find the solution. If you are still unsure, you can try to use other methods such as elimination or matrices to find the solution.

Conclusion

In conclusion, determining if a point is a solution to a system of equations is an essential skill in mathematics. By understanding how to verify solutions, you can gain a deeper understanding of the system and its behavior, and make informed decisions based on the results.

Final Thoughts

The concept of checking solutions to a system of equations is a fundamental aspect of mathematics, and it requires a deep understanding of algebra and geometry. By mastering this skill, you can develop a strong foundation in mathematics and apply it to real-world problems and applications.

References

• [1] "Algebra and Geometry" by Michael Artin
• [2] "Linear Algebra and Its Applications" by Gilbert Strang
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Additional Resources

• Khan Academy: Systems of Linear Equations
• MIT OpenCourseWare: Linear Algebra
• Wolfram Alpha: Systems of Linear Equations