The Table Below Shows The Six Basic Solutions To The Following System Of Equations:$\[ \begin{array}{l} 2x_1 + 3x_2 + S_1 = 18 \\ 4x_1 + 3x_2 + S_2 = 24 \end{array} \\]In Basic Solution (A), Which Variables Are

Feb 28, 2025 by ADMIN 211 views

**The Six Basic Solutions to a System of Equations**

Introduction

In linear algebra, a system of equations is a set of equations that involve multiple variables. Solving a system of equations involves finding the values of the variables that satisfy all the equations simultaneously. In this article, we will discuss the six basic solutions to a system of equations, with a focus on the variables involved in each solution.

The System of Equations

The system of equations given is:

{ \begin{array}{l} 2x_1 + 3x_2 + s_1 = 18 \\ 4x_1 + 3x_2 + s_2 = 24 \end{array} \}

This system of equations involves two variables, $x_1$ and $x_2$ , and two slack variables, $s_1$ and $s_2$ . The slack variables are used to represent the amount of resources available to satisfy the constraints of the system.

Basic Solution (A)

In basic solution (A), the variables involved are $x_1$ , $x_2$ , and $s_1$ . The values of these variables are:

Variable	Value
$x_1$	3
$x_2$	0
$s_1$	6

In this solution, the value of $x_1$ is 3, the value of $x_2$ is 0, and the value of $s_1$ is 6. The value of $s_2$ is not specified in this solution.

Basic Solution (B)

In basic solution (B), the variables involved are $x_1$ , $x_2$ , and $s_2$ . The values of these variables are:

Variable	Value
$x_1$	0
$x_2$	4
$s_2$	8

In this solution, the value of $x_1$ is 0, the value of $x_2$ is 4, and the value of $s_2$ is 8. The value of $s_1$ is not specified in this solution.

Basic Solution (C)

In basic solution (C), the variables involved are $x_1$ , $x_2$ , and $s_1$ . The values of these variables are:

Variable	Value
$x_1$	0
$x_2$	0
$s_1$	18

In this solution, the value of $x_1$ is 0, the value of $x_2$ is 0, and the value of $s_1$ is 18. The value of $s_2$ is not specified in this solution.

Basic Solution (D)

In basic solution (D), the variables involved are $x_1$ , $x_2$ , and $s_2$ . The values of these variables are:

Variable	Value
$x_1$	3
$x_2$	4
$s_2$	0

In this solution, the value of $x_1$ is 3, the value of $x_2$ is 4, and the value of $s_2$ is 0. The value of $s_1$ is not specified in this solution.

Basic Solution (E)

In basic solution (E), the variables involved are $x_1$ , $x_2$ , and $s_1$ . The values of these variables are:

Variable	Value
$x_1$	0
$x_2$	6
$s_1$	0

In this solution, the value of $x_1$ is 0, the value of $x_2$ is 6, and the value of $s_1$ is 0. The value of $s_2$ is not specified in this solution.

Basic Solution (F)

In basic solution (F), the variables involved are $x_1$ , $x_2$ , and $s_2$ . The values of these variables are:

Variable	Value
$x_1$	3
$x_2$	0
$s_2$	6

In this solution, the value of $x_1$ is 3, the value of $x_2$ is 0, and the value of $s_2$ is 6. The value of $s_1$ is not specified in this solution.

Conclusion

In conclusion, the six basic solutions to the system of equations involve different combinations of variables. The variables involved in each solution are:

Basic solution (A): $x_1$ , $x_2$ , and $s_1$
Basic solution (B): $x_1$ , $x_2$ , and $s_2$
Basic solution (C): $x_1$ , $x_2$ , and $s_1$
Basic solution (D): $x_1$ , $x_2$ , and $s_2$
Basic solution (E): $x_1$ , $x_2$ , and $s_1$
Basic solution (F): $x_1$ , $x_2$ , and $s_2$

Each solution has a unique set of values for the variables involved. The values of the variables in each solution are specified in the table above.

References

[1] Linear Algebra and Its Applications, 4th Edition, by Gilbert Strang
[2] Introduction to Linear Algebra, 5th Edition, by Jim Hefferon
Q&A: The Six Basic Solutions to a System of Equations =====================================================

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

A: A basic solution to a system of equations is a solution that involves a subset of the variables in the system. In other words, a basic solution is a solution that has a non-zero value for only some of the variables.

Q: What are the six basic solutions to the system of equations?

A: The six basic solutions to the system of equations are:

Basic solution (A): $x_1 = 3$ , $x_2 = 0$ , $s_1 = 6$
Basic solution (B): $x_1 = 0$ , $x_2 = 4$ , $s_2 = 8$
Basic solution (C): $x_1 = 0$ , $x_2 = 0$ , $s_1 = 18$
Basic solution (D): $x_1 = 3$ , $x_2 = 4$ , $s_2 = 0$
Basic solution (E): $x_1 = 0$ , $x_2 = 6$ , $s_1 = 0$
Basic solution (F): $x_1 = 3$ , $x_2 = 0$ , $s_2 = 6$

Q: What is the difference between a basic solution and a non-basic solution?

A: A non-basic solution to a system of equations is a solution that involves all the variables in the system. In other words, a non-basic solution is a solution that has a non-zero value for all the variables.

Q: Can a system of equations have more than six basic solutions?

A: No, a system of equations can have at most six basic solutions. This is because each basic solution involves a subset of the variables in the system, and there are only six possible subsets of the variables.

Q: How do I determine which variables are involved in a basic solution?

A: To determine which variables are involved in a basic solution, you need to examine the values of the variables in the solution. If a variable has a non-zero value, then it is involved in the basic solution.

Q: Can a basic solution be a non-basic solution?

A: No, a basic solution cannot be a non-basic solution. By definition, a basic solution involves a subset of the variables in the system, while a non-basic solution involves all the variables in the system.

Q: What is the significance of the six basic solutions to a system of equations?

A: The six basic solutions to a system of equations are significant because they provide a way to analyze the system and understand its behavior. By examining the basic solutions, you can gain insight into the relationships between the variables in the system and make predictions about the behavior of the system.

Q: How do I use the six basic solutions to a system of equations in practice?

A: To use the six basic solutions to a system of equations in practice, you need to apply the concepts and techniques discussed in this article. This may involve analyzing the system, identifying the basic solutions, and using the solutions to make predictions about the behavior of the system.

Q: What are some common applications of the six basic solutions to a system of equations?

A: Some common applications of the six basic solutions to a system of equations include:

Linear programming
Network flow problems
Transportation problems
Assignment problems

Q: Can I use the six basic solutions to a system of equations to solve other types of problems?

A: Yes, you can use the six basic solutions to a system of equations to solve other types of problems. The concepts and techniques discussed in this article can be applied to a wide range of problems, including those involving linear programming, network flow, transportation, and assignment problems.