Select The Correct Answer.Which System Of Equations Has A Solution Of \[$(1, 3, -1)\$\]?A. $\[ \begin{aligned} x + 2y &= 7 \\ y + 2z &= 1 \\ x - Y - Z &= -1 \end{aligned} \\]B.$\[ \begin{aligned} x + Y &= 4 \\ y - Z &= 2 \\ x + Y -

Feb 28, 2025 by ADMIN 232 views

**System of Equations: Selecting the Correct Answer**

Introduction

In mathematics, a system of equations is a set of equations that are all true at the same time. Solving a system of equations involves finding the values of the variables that make all the equations true. In this article, we will explore which system of equations has a solution of $(1, 3, -1)$ .

System of Equations A

The first system of equations is given by:

{ \begin{aligned} x + 2y &= 7 \\ y + 2z &= 1 \\ x - y - z &= -1 \end{aligned} \}

To determine if this system has a solution of $(1, 3, -1)$ , we need to substitute these values into each equation and check if they are true.

Substituting Values into System A

Let's substitute $x = 1$ , $y = 3$ , and $z = -1$ into each equation:

$x + 2y = 7$ $1 + 2(3) = 7$ $1 + 6 = 7$ $7 = 7$ (True)
$y + 2z = 1$ $3 + 2(-1) = 1$ $3 - 2 = 1$ $1 = 1$ (True)
$x - y - z = -1$ $1 - 3 - (-1) = -1$ $1 - 3 + 1 = -1$ $-1 = -1$ (True)

System of Equations B

The second system of equations is given by:

{ \begin{aligned} x + y &= 4 \\ y - z &= 2 \\ x + y - z &= -1 \end{aligned} \}

To determine if this system has a solution of $(1, 3, -1)$ , we need to substitute these values into each equation and check if they are true.

Substituting Values into System B

Let's substitute $x = 1$ , $y = 3$ , and $z = -1$ into each equation:

$x + y = 4$ $1 + 3 = 4$ $4 = 4$ (True)
$y - z = 2$ $3 - (-1) = 2$ $3 + 1 = 2$ $4 = 2$ (False)

Since the second equation is false, we can conclude that System B does not have a solution of $(1, 3, -1)$ .

Conclusion

Based on our analysis, we can conclude that System A has a solution of $(1, 3, -1)$ , while System B does not.

Why is System A the Correct Answer?

System A is the correct answer because all three equations in the system are true when $x = 1$ , $y = 3$ , and $z = -1$ . This means that the values of $x$ , $y$ , and $z$ satisfy all three equations simultaneously, making System A a consistent system of equations.

What is the Significance of System A?

System A is significant because it demonstrates that a system of equations can have a solution that satisfies all the equations simultaneously. This is an important concept in mathematics, as it allows us to solve systems of equations and find the values of the variables that make all the equations true.

What are the Implications of System A?

The implications of System A are that it provides a framework for solving systems of equations. By analyzing the equations and finding the values of the variables that satisfy all the equations simultaneously, we can solve systems of equations and find the solutions.

What are the Applications of System A?

The applications of System A are numerous. It can be used to solve problems in physics, engineering, economics, and other fields where systems of equations are used to model real-world phenomena. By applying the concepts and techniques learned from System A, we can solve complex problems and find the solutions.

What are the Limitations of System A?

The limitations of System A are that it assumes that the system of equations has a solution. In some cases, a system of equations may not have a solution, or it may have multiple solutions. In such cases, System A may not be applicable.

What are the Future Directions of System A?

The future directions of System A are to explore its applications in different fields and to develop new techniques and methods for solving systems of equations. By continuing to research and develop new methods, we can improve our ability to solve systems of equations and find the solutions.

Conclusion

In conclusion, System A is the correct answer because it has a solution of $(1, 3, -1)$ . This system demonstrates that a system of equations can have a solution that satisfies all the equations simultaneously. The implications of System A are that it provides a framework for solving systems of equations and has numerous applications in different fields.

Introduction

In our previous article, we explored which system of equations has a solution of $(1, 3, -1)$ . We found that System A has a solution of $(1, 3, -1)$ , while System B does not. In this article, we will answer some frequently asked questions about system of equations.

Q: What is a system of equations?

A: A system of equations is a set of equations that are all true at the same time. Solving a system of equations involves finding the values of the variables that make all the equations true.

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

A: To determine if a system of equations has a solution, you need to substitute the values of the variables into each equation and check if they are true. If all the equations are true, then the system has a solution.

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

A: A single equation is an equation that has only one variable. A system of equations, on the other hand, is a set of equations that have multiple variables. Solving a single equation involves finding the value of one variable, while solving a system of equations involves finding the values of multiple variables.

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 multiple sets of values that satisfy all the equations.

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to use a method such as substitution or elimination. Substitution involves substituting the values of one variable into another equation, while elimination involves adding or subtracting equations to eliminate one variable.

Q: What are some common methods for solving systems of equations?

A: Some common methods for solving systems of equations include substitution, elimination, and graphing. Substitution involves substituting the values of one variable into another equation, while elimination involves adding or subtracting equations to eliminate one variable. Graphing involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: Can a system of equations have no solution?

A: Yes, a system of equations can have no solution. This occurs when the system has no values that satisfy all the equations.

Q: What are some real-world applications of systems of equations?

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

Q: How do I know if a system of equations is consistent or inconsistent?

A: A system of equations is consistent if it has a solution, while an inconsistent system has no solution. To determine if a system is consistent or inconsistent, you need to substitute the values of the variables into each equation and check if they are true.

Q: What is the significance of systems of equations in mathematics?

A: Systems of equations are an important concept in mathematics, as they provide a framework for solving equations with multiple variables. They are used to model real-world phenomena and have numerous applications in different fields.

Q: Can a system of equations have an infinite number of solutions?

A: Yes, a system of equations can have an infinite number of solutions. This occurs when the system has an equation that is true for all values of the variables.

Conclusion

In conclusion, systems of equations are an important concept in mathematics, as they provide a framework for solving equations with multiple variables. They have numerous real-world applications and are used to model real-world phenomena. By understanding systems of equations, you can solve complex problems and find the solutions.

Frequently Asked Questions

Q: What is a system of equations?
A: A system of equations is a set of equations that are all true at the same time.
Q: How do I know if a system of equations has a solution?
A: To determine if a system of equations has a solution, you need to substitute the values of the variables into each equation and check if they are true.
Q: What is the difference between a system of equations and a single equation?
A: A single equation is an equation that has only one variable, while a system of equations is a set of equations that have multiple variables.
Q: Can a system of equations have multiple solutions?
A: Yes, a system of equations can have multiple solutions.
Q: How do I solve a system of equations?
A: To solve a system of equations, you need to use a method such as substitution or elimination.

Glossary

System of equations: A set of equations that are all true at the same time.
Consistent system: A system of equations that has a solution.
Inconsistent system: A system of equations that has no solution.
Substitution: A method for solving systems of equations by substituting the values of one variable into another equation.
Elimination: A method for solving systems of equations by adding or subtracting equations to eliminate one variable.
Graphing: A method for solving systems of equations by graphing the equations on a coordinate plane and finding the point of intersection.