Is { (2,-7)$}$ A Solution To This System Of Equations?${ \begin{array}{l} 5x + 2y = 7 \ 13x + 5y = -9 \end{array} }$A. YesB. No

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Introduction

When dealing with systems of linear equations, it's essential to determine whether a given point satisfies both equations. In this case, we're presented with a system of two linear equations and a point {(2,-7)$}$. Our goal is to determine whether this point is a solution to the system of equations.

Understanding the System of Equations

The given system of equations consists of two linear equations:

1. $5x + 2y = 7$
2. $13x + 5y = -9$

To determine whether the point {(2,-7)$}$ is a solution, we need to substitute the values of $x$ and $y$ from the point into both equations and check if the resulting statements are true.

Substituting the Point into the First Equation

Let's start by substituting the values of $x$ and $y$ from the point {(2,-7)$}$ into the first equation:

$5x + 2y = 7$

Substituting $x = 2$ and $y = -7$, we get:

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

Expanding and simplifying the equation, we get:

$10 - 14 = 7$

This simplifies to:

$-4 = 7$

As we can see, the resulting statement is false. This means that the point {(2,-7)$}$ does not satisfy the first equation.

Substituting the Point into the Second Equation

Now, let's substitute the values of $x$ and $y$ from the point {(2,-7)$}$ into the second equation:

$13x + 5y = -9$

Substituting $x = 2$ and $y = -7$, we get:

$13(2) + 5(-7) = -9$

Expanding and simplifying the equation, we get:

$26 - 35 = -9$

This simplifies to:

$-9 = -9$

As we can see, the resulting statement is true. This means that the point {(2,-7)$}$ satisfies the second equation.

Conclusion

Based on our analysis, we can conclude that the point {(2,-7)$}$ does not satisfy the first equation, but it does satisfy the second equation. Since a solution to a system of equations must satisfy both equations, we can conclude that the point {(2,-7)$}$ is not a solution to the given system of equations.

Final Answer

The final answer is: B. No

Discussion

In this discussion, we've seen how to determine whether a given point is a solution to a system of linear equations. We've also seen how to substitute the values of $x$ and $y$ from the point into both equations and check if the resulting statements are true. By following these steps, we can determine whether a point is a solution to a system of equations.

Related Topics

• Systems of linear equations
• Substitution method
• Linear equations
• Algebra

References

• [1] "Systems of Linear Equations" by Math Open Reference
• [2] "Substitution Method" by Khan Academy
• [3] "Linear Equations" by Purplemath

Additional Resources

• [1] "Systems of Linear Equations" by MIT OpenCourseWare
• [2] "Substitution Method" by IXL
• [3] "Linear Equations" by Mathway
  

Introduction

In our previous article, we discussed how to determine whether a given point is a solution to a system of linear equations. In this article, we'll answer some frequently asked questions about systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. Each equation in the system is a linear equation, which means it can be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

A: There are several methods to solve a system of linear equations, including the substitution method, the elimination method, and the graphing method. The substitution method involves substituting the value of one variable from one equation into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables. The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the substitution method?

A: The substitution method is a method of solving a system of linear equations by substituting the value of one variable from one equation into the other equation. This method is useful when one of the equations is easily solvable for one of the variables.

Q: What is the elimination method?

A: The elimination method is a method of solving a system of linear equations by 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 graphing method?

A: The graphing method is a method of solving a system of linear equations by graphing the equations on a coordinate plane and finding the point of intersection. This method is useful when the equations are easily graphable.

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

A: To determine whether a point is a solution to a system of linear equations, you need to substitute the values of x and y from the point into both equations and check if the resulting statements are true. If the point satisfies both equations, then it is a solution to the system.

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

A: A system of linear equations consists of two or more linear equations, while a system of nonlinear equations consists of two or more nonlinear equations. Nonlinear equations are equations that cannot be written in the form ax + by = c, where a, b, and c are constants, and x and y are variables.

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

A: Solving a system of nonlinear equations can be more challenging than solving a system of linear equations. There are several methods to solve a system of nonlinear equations, including the substitution method, the elimination method, and numerical methods. The substitution method involves substituting the value of one variable from one equation into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables. Numerical methods involve using numerical techniques, such as the Newton-Raphson method, to approximate the solution.

Q: What is the importance of systems of linear equations in real-life applications?

A: Systems of linear equations have numerous applications in real-life situations, such as physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the flow of fluids, and the behavior of electrical circuits.

Q: How do I use systems of linear equations in real-life applications?

A: To use systems of linear equations in real-life applications, you need to identify the variables and the equations that describe the problem. You then need to solve the system of equations to find the values of the variables. This can be done using the substitution method, the elimination method, or numerical methods.

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:

• Not checking if the equations are linear
• Not using the correct method to solve the system
• Not checking if the solution satisfies both equations
• Not considering the possibility of multiple solutions or no solution

Q: How do I check if a solution satisfies both equations?

A: To check if a solution satisfies both equations, you need to substitute the values of x and y from the solution into both equations and check if the resulting statements are true. If the solution satisfies both equations, then it is a valid solution.

Q: What are some common applications of systems of linear equations?

A: Some common applications of systems of linear equations include:

• Physics: modeling the motion of objects
• Engineering: designing electrical circuits and mechanical systems
• Economics: modeling the behavior of economic systems
• Computer science: solving problems in computer graphics and game development

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

A: To use systems of linear equations to model real-world problems, you need to identify the variables and the equations that describe the problem. You then need to solve the system of equations to find the values of the variables. This can be done using the substitution method, the elimination method, or numerical methods.

Q: What are some common challenges when solving systems of linear equations?

A: Some common challenges when solving systems of linear equations include:

• Dealing with multiple solutions or no solution
• Dealing with complex or nonlinear equations
• Dealing with large or sparse systems of equations
• Dealing with numerical instability or round-off errors

Q: How do I overcome these challenges?

A: To overcome these challenges, you need to use the correct method to solve the system, check if the solution satisfies both equations, and consider the possibility of multiple solutions or no solution. You also need to use numerical methods or approximation techniques to deal with complex or nonlinear equations.

Q: What are some common tools and software used to solve systems of linear equations?

A: Some common tools and software used to solve systems of linear equations include:

• MATLAB
• Mathematica
• Maple
• Python libraries such as NumPy and SciPy
• C++ libraries such as Eigen and Armadillo

Q: How do I choose the right tool or software for solving systems of linear equations?

A: To choose the right tool or software for solving systems of linear equations, you need to consider the size and complexity of the system, the type of equations, and the desired level of accuracy. You also need to consider the availability of resources, the cost, and the ease of use.

Q: What are some common applications of systems of linear equations in science and engineering?

A: Some common applications of systems of linear equations in science and engineering include:

• Physics: modeling the motion of objects, the behavior of electrical circuits, and the behavior of mechanical systems
• Engineering: designing electrical circuits, mechanical systems, and computer networks
• Computer science: solving problems in computer graphics, game development, and data analysis

Q: How do I use systems of linear equations to solve problems in science and engineering?

A: To use systems of linear equations to solve problems in science and engineering, you need to identify the variables and the equations that describe the problem. You then need to solve the system of equations to find the values of the variables. This can be done using the substitution method, the elimination method, or numerical methods.

Q: What are some common challenges when using systems of linear equations in science and engineering?

A: Some common challenges when using systems of linear equations in science and engineering include:

• Dealing with complex or nonlinear equations
• Dealing with large or sparse systems of equations
• Dealing with numerical instability or round-off errors
• Dealing with the need for high accuracy and precision

Q: How do I overcome these challenges?

A: To overcome these challenges, you need to use the correct method to solve the system, check if the solution satisfies both equations, and consider the possibility of multiple solutions or no solution. You also need to use numerical methods or approximation techniques to deal with complex or nonlinear equations.

Q: What are some common applications of systems of linear equations in economics?

A: Some common applications of systems of linear equations in economics include:

• Modeling the behavior of economic systems
• Analyzing the impact of policy changes on the economy
• Forecasting economic trends and patterns
• Optimizing economic outcomes

Q: How do I use systems of linear equations to solve problems in economics?

A: To use systems of linear equations to solve problems in economics, you need to identify the variables and the equations that describe the problem. You then need to solve the system of equations to find the values of the variables. This can be done using the substitution method, the elimination method, or numerical methods.

Q: What are some common challenges when using systems of linear equations in economics?

A: Some common challenges when using systems of linear equations in economics include:

• Dealing with complex or nonlinear equations
• Dealing with large or sparse systems of equations
• Dealing with numerical instability or round-off errors
• Dealing with the need for high accuracy and precision

Q: How do I overcome these challenges?

A: To overcome these challenges, you need to use the correct method to solve the system, check if the solution satisfies both equations, and consider the possibility of multiple solutions or no solution. You also need to use numerical methods or approximation techniques to deal with complex or nonlinear equations.

Q: What are some common applications of systems of linear equations in computer science?

A: Some common applications of systems of linear equations in computer science include:

• Solving problems in computer graphics and game development
• Analyzing the behavior of computer networks
• Optimizing algorithms and data structures