The Table Below Shows The Six Basic Solutions To The Following System Of Equations:${ \begin{array}{l} 2 X_1 + 3 X_2 + S_1 = 27 \ 4 X_1 + 3 X_2 + S_2 = 36 \end{array} }$In Basic Solution (B), Which Variables Are

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 the basic solution (B).

The System of Equations

The system of equations given is:

{ \begin{array}{l} 2 x_1 + 3 x_2 + s_1 = 27 \\ 4 x_1 + 3 x_2 + s_2 = 36 \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.

The Six Basic Solutions

The six basic solutions to the system of equations are shown in the table below:

Variables in Basic Solution (B)

In basic solution (B), the variables involved are $x_2$ and $s_2$. The value of $x_2$ is 9, and the value of $s_2$ is 0. This means that the variable $x_2$ is at its maximum value, and the variable $s_2$ is at its minimum value.

Why $x_2$ is at its maximum value

In basic solution (B), the value of $x_2$ is 9. This is because the second equation in the system of equations is $4 x_1 + 3 x_2 + s_2 = 36$. Since the value of $s_2$ is 0, the value of $x_2$ must be at its maximum value to satisfy the equation.

Why $s_2$ is at its minimum value

In basic solution (B), the value of $s_2$ is 0. This is because the second equation in the system of equations is $4 x_1 + 3 x_2 + s_2 = 36$. Since the value of $x_2$ is 9, the value of $s_2$ must be at its minimum value to satisfy the equation.

Conclusion

In this article, we discussed the six basic solutions to a system of equations. We focused on the variables involved in basic solution (B) and explained why $x_2$ is at its maximum value and $s_2$ is at its minimum value. Understanding the basic solutions to a system of equations is crucial in linear algebra and has many practical applications in fields such as economics, engineering, and computer science.

References

• [1] Hillier, F. S., & Lieberman, G. J. (2015). Introduction to operations research. McGraw-Hill Education.
• [2] Winston, W. L. (2014). Operations research: Applications and algorithms. Cengage Learning.

Further Reading

• [1] Linear Algebra and Its Applications by Gilbert Strang
• [2] Introduction to Linear Algebra by Gilbert Strang
• [3] Operations Research: An Introduction by Frederick S. Hillier and Gerald J. Lieberman
  The Six Basic Solutions to a System of Equations: Q&A =====================================================

Introduction

In our previous article, we discussed the six basic solutions to a system of equations. In this article, we will answer some frequently asked questions about the six basic solutions and provide additional insights into the topic.

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

A: A basic solution to a system of equations is a solution in which all the variables are at their minimum or maximum values. In other words, a basic solution is a solution in which all the variables are either at their minimum value (0) or at their maximum value.

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

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

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

A: A basic solution is a solution in which all the variables are at their minimum or maximum values. A non-basic solution, on the other hand, is a solution in which at least one variable is not at its minimum or maximum value.

Q: Why are basic solutions important in linear algebra?

A: Basic solutions are important in linear algebra because they provide a way to solve systems of equations. By finding the basic solutions to a system of equations, we can determine the values of the variables that satisfy the equations.

Q: How do I find the basic solutions to a system of equations?

A: To find the basic solutions to a system of equations, you can use the following steps:

1. Write the system of equations in standard form.
2. Find the basic feasible solutions by setting all the variables to their minimum or maximum values.
3. Check if the basic feasible solutions satisfy the equations.

Q: What are some common applications of basic solutions in real-world problems?

A: Basic solutions have many practical applications in fields such as economics, engineering, and computer science. Some common applications of basic solutions include:

• Linear programming: Basic solutions are used to solve linear programming problems, which involve finding the optimal solution to a system of linear equations.
• Network flow problems: Basic solutions are used to solve network flow problems, which involve finding the maximum or minimum flow in a network.
• Scheduling problems: Basic solutions are used to solve scheduling problems, which involve finding the optimal schedule for a set of tasks.

Conclusion

In this article, we answered some frequently asked questions about the six basic solutions to a system of equations and provided additional insights into the topic. We hope that this article has been helpful in understanding the concept of basic solutions and their importance in linear algebra.

References

• [1] Hillier, F. S., & Lieberman, G. J. (2015). Introduction to operations research. McGraw-Hill Education.
• [2] Winston, W. L. (2014). Operations research: Applications and algorithms. Cengage Learning.

Further Reading

• [1] Linear Algebra and Its Applications by Gilbert Strang
• [2] Introduction to Linear Algebra by Gilbert Strang
• [3] Operations Research: An Introduction by Frederick S. Hillier and Gerald J. Lieberman