The Solution Set Is \[$\langle(\square, \square, \square)\rangle\$\].Simplify Your Answers.$\[ \left[\begin{array}{rrr|r} 1 & 5 & 0 & -4 \\ 0 & 1 & -\frac{3}{20} & -\frac{13}{10} \\ 0 & 0 & 1 &

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Introduction

In mathematics, particularly in linear algebra, the solution set of a system of equations is a crucial concept that helps us understand the relationship between the variables and the equations. The solution set is the set of all possible solutions that satisfy the system of equations. In this article, we will delve into the solution set of a given system of equations and simplify our answers.

The System of Equations

The given system of equations is:

$\left[\begin{array}{rrr|r} 1 & 5 & 0 & -4 \\ 0 & 1 & -\frac{3}{20} & -\frac{13}{10} \\ 0 & 0 & 1 & \end{array}\right]$

This system of equations consists of three equations with three variables. The first equation is $x + 5y + 0z = -4$, the second equation is $0x + 1y - \frac{3}{20}z = -\frac{13}{10}$, and the third equation is $0x + 0y + 1z = 1$.

Simplifying the System of Equations

To simplify the system of equations, we can use the method of substitution or elimination. In this case, we will use the method of substitution. We will start by solving the third equation for $z$, which gives us $z = 1$.

Substituting $z$ into the Second Equation

Now that we have the value of $z$, we can substitute it into the second equation. This gives us:

$0x + 1y - \frac{3}{20}(1) = -\frac{13}{10}$

Simplifying this equation, we get:

$y - \frac{3}{20} = -\frac{13}{10}$

Solving for $y$

To solve for $y$, we can add $\frac{3}{20}$ to both sides of the equation. This gives us:

$y = -\frac{13}{10} + \frac{3}{20}$

Simplifying this expression, we get:

$y = -\frac{26}{20} + \frac{3}{20}$

$y = -\frac{23}{20}$

Substituting $y$ into the First Equation

Now that we have the value of $y$, we can substitute it into the first equation. This gives us:

$x + 5(-\frac{23}{20}) + 0z = -4$

Simplifying this equation, we get:

$x - \frac{115}{20} = -4$

Solving for $x$

To solve for $x$, we can add $\frac{115}{20}$ to both sides of the equation. This gives us:

$x = -4 + \frac{115}{20}$

Simplifying this expression, we get:

$x = -\frac{80}{20} + \frac{115}{20}$

$x = \frac{35}{20}$

$x = \frac{7}{4}$

The Solution Set

The solution set of the system of equations is the set of all possible solutions that satisfy the system of equations. In this case, the solution set is:

$\langle(\frac{7}{4}, -\frac{23}{20}, 1)\rangle$

This means that the only solution to the system of equations is $x = \frac{7}{4}$, $y = -\frac{23}{20}$, and $z = 1$.

Conclusion

In conclusion, the solution set of the given system of equations is $\langle(\frac{7}{4}, -\frac{23}{20}, 1)\rangle$. This means that the only solution to the system of equations is $x = \frac{7}{4}$, $y = -\frac{23}{20}$, and $z = 1$. We simplified the system of equations using the method of substitution and solved for the values of $x$, $y$, and $z$. The solution set is a crucial concept in mathematics, particularly in linear algebra, and is used to understand the relationship between the variables and the equations.

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Introduction

In our previous article, we delved into the solution set of a given system of equations and simplified our answers. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information on the solution set.

Q: What is the solution set of a system of equations?

A: The solution set of a system of equations is the set of all possible solutions that satisfy the system of equations. It is a crucial concept in mathematics, particularly in linear algebra, and is used to understand the relationship between the variables and the equations.

Q: How do I find the solution set of a system of equations?

A: To find the solution set of a system of equations, you can use the method of substitution or elimination. In our previous article, we used the method of substitution to simplify the system of equations and solve for the values of $x$, $y$, and $z$.

Q: What is the difference between the solution set and the general solution?

A: The solution set and the general solution are two related but distinct concepts. The solution set is the set of all possible solutions that satisfy the system of equations, while the general solution is a specific solution that satisfies the system of equations. In our previous article, we found the solution set to be $\langle(\frac{7}{4}, -\frac{23}{20}, 1)\rangle$, which means that the only solution to the system of equations is $x = \frac{7}{4}$, $y = -\frac{23}{20}$, and $z = 1$.

Q: Can I have multiple solutions to a system of equations?

A: Yes, it is possible to have multiple solutions to a system of equations. However, in our previous article, we found that the solution set is a single solution, which means that there is only one possible solution that satisfies the system of equations.

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 can use the method of substitution or elimination. If the system of equations has a solution, then the solution set will be a non-empty set. In our previous article, we found that the solution set is non-empty, which means that the system of equations has a solution.

Q: Can I use the solution set to solve other systems of equations?

A: Yes, the solution set can be used to solve other systems of equations. If you have a system of equations that is similar to the one we solved in our previous article, then you can use the solution set to find the solution to the new system of equations.

Q: What are some common applications of the solution set?

A: The solution set has many applications in mathematics, science, and engineering. Some common applications include:

• Solving systems of linear equations
• Finding the intersection of two or more lines or planes
• Determining the stability of a system
• Modeling real-world phenomena

Conclusion

In conclusion, the solution set is a crucial concept in mathematics, particularly in linear algebra, and is used to understand the relationship between the variables and the equations. We provided a Q&A section to further clarify any doubts and provide additional information on the solution set. We hope that this article has been helpful in understanding the solution set and its applications.