Betty Correctly Determined That The Ordered Pair \[$(-3,5)\$\] Is A Solution To The System Of Linear Equations \[$6x + 5y = 7\$\] And \[$x + 4y = 17\$\].Based On This Information, Which Statement Is Correct?A.

Introduction

In mathematics, solving systems of linear equations is a fundamental concept that involves finding the solution to a set of linear equations. A system of linear equations is a collection of two or more linear equations that are solved simultaneously to find the values of the variables. In this article, we will explore how to solve systems of linear equations using the given information about the ordered pair ${(-3,5)}$ being a solution to the system of linear equations ${6x + 5y = 7}$ and ${x + 4y = 17}$.

Understanding the Problem

The problem states that the ordered pair ${(-3,5)}$ is a solution to the system of linear equations ${6x + 5y = 7}$ and ${x + 4y = 17}$. This means that when we substitute the values of x and y from the ordered pair into the two equations, both equations should be satisfied. Our goal is to determine which statement is correct based on this information.

Substituting the Ordered Pair into the Equations

To verify that the ordered pair ${(-3,5)}$ is a solution to the system of linear equations, we need to substitute the values of x and y into both equations.

Substituting into the First Equation

Substituting x = -3 and y = 5 into the first equation ${6x + 5y = 7}$, we get:

${6(-3) + 5(5) = 7}$

Expanding and simplifying the equation, we get:

${-18 + 25 = 7}$

Simplifying further, we get:

${7 = 7}$

This confirms that the ordered pair ${(-3,5)}$ satisfies the first equation.

Substituting into the Second Equation

Substituting x = -3 and y = 5 into the second equation ${x + 4y = 17}$, we get:

${(-3) + 4(5) = 17}$

Expanding and simplifying the equation, we get:

${-3 + 20 = 17}$

Simplifying further, we get:

${17 = 17}$

This confirms that the ordered pair ${(-3,5)}$ satisfies the second equation.

Determining the Correct Statement

Based on the information provided, we can determine which statement is correct. The correct statement is:

A. The ordered pair ${(-3,5)}$ is a solution to the system of linear equations ${6x + 5y = 7}$ and ${x + 4y = 17}$.

Conclusion

In conclusion, we have verified that the ordered pair ${(-3,5)}$ is a solution to the system of linear equations ${6x + 5y = 7}$ and ${x + 4y = 17}$. This confirms that the statement A. is correct.

Additional Tips and Tricks

When solving systems of linear equations, it's essential to:

• Verify that the ordered pair satisfies both equations.
• Use substitution or elimination methods to solve the system.
• Check for any inconsistencies or contradictions in the system.

By following these tips and tricks, you can become proficient in solving systems of linear equations and apply this knowledge to a wide range of mathematical and real-world problems.

Common Mistakes to Avoid

When solving systems of linear equations, it's essential to avoid common mistakes such as:

• Failing to verify that the ordered pair satisfies both equations.
• Using incorrect methods or techniques to solve the system.
• Ignoring any inconsistencies or contradictions in the system.

By being aware of these common mistakes, you can avoid them and ensure that your solutions are accurate and reliable.

Real-World Applications

Solving systems of linear equations has numerous real-world applications in fields such as:

• Physics and engineering: Solving systems of linear equations is essential in modeling and analyzing physical systems, such as electrical circuits and mechanical systems.
• Computer science: Solving systems of linear equations is used in computer graphics, machine learning, and data analysis.
• Economics: Solving systems of linear equations is used in modeling economic systems, such as supply and demand curves.

By understanding how to solve systems of linear equations, you can apply this knowledge to a wide range of mathematical and real-world problems.

Conclusion

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Introduction

In our previous article, we explored how to solve systems of linear equations using the given information about the ordered pair ${(-3,5)}$ being a solution to the system of linear equations ${6x + 5y = 7}$ and ${x + 4y = 17}$. In this article, we will provide a Q&A guide to help you better understand how to solve systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a collection of two or more linear equations that are solved simultaneously to find the values of the variables.

Q: How do I know if a system of linear equations has a solution?

A: To determine if a system of linear equations has a solution, you need to check if the equations are consistent. If the equations are consistent, then the system has a solution. If the equations are inconsistent, then the system does not have a solution.

Q: What is the difference between a consistent and inconsistent system of linear equations?

A: A consistent system of linear equations is one where the equations have a solution. An inconsistent system of linear equations is one where the equations do not have a solution.

Q: How do I solve a system of linear equations?

A: There are several methods to solve a system of linear equations, including:

• Substitution method: This involves substituting the values of one variable into the other equation to solve for the other variable.
• Elimination method: This involves adding or subtracting the equations to eliminate one of the variables.
• Graphical method: This involves graphing the equations on a coordinate plane to find the point of intersection.

Q: What is the substitution method?

A: The substitution method involves substituting the values of one variable into the other equation to solve for the other variable. This method is useful when one of the equations is already solved for one of the variables.

Q: What is the elimination method?

A: The elimination method involves adding or subtracting the equations to eliminate one of the variables. This method is useful when the coefficients of one of the variables are the same in both equations.

Q: What is the graphical method?

A: The graphical method involves graphing the equations on a coordinate plane to find the point of intersection. This method is useful when the equations are linear and the point of intersection is easy to find.

Q: How do I verify that the solution is correct?

A: To verify that the solution is correct, you need to substitute the values of the variables back into the original equations to check if they are satisfied.

Q: What are some common mistakes to avoid when solving systems of linear equations?

A: Some common mistakes to avoid when solving systems of linear equations include:

• Failing to verify that the solution is correct.
• Using incorrect methods or techniques to solve the system.
• Ignoring any inconsistencies or contradictions in the system.

Q: How do I apply systems of linear equations to real-world problems?

A: Systems of linear equations have numerous real-world applications in fields such as physics, engineering, computer science, and economics. By understanding how to solve systems of linear equations, you can apply this knowledge to a wide range of mathematical and real-world problems.

Conclusion

In conclusion, solving systems of linear equations is a fundamental concept in mathematics that involves finding the solution to a set of linear equations. By following the steps outlined in this article, you can better understand how to solve systems of linear equations and apply this knowledge to a wide range of mathematical and real-world problems.