Consider The System:${ \begin{array}{l} y = 3x + 5 \ y = Ax + B \end{array} }$1. What Values For A A A And B B B Make The System Inconsistent?2. What Values For A A A And B B B Make The System Consistent And

Introduction

In mathematics, a system of linear equations is a set of two or more equations that involve two or more variables. Solving a system of linear equations involves finding the values of the variables that satisfy all the equations in the system. In this article, we will consider a system of two linear equations and explore the conditions under which the system is inconsistent and consistent.

The System of Linear Equations

The system of linear equations we will consider is:

$$\begin{array}{l} y = 3x + 5 \\ y = ax + b \end{array}$$

where $a$ and $b$ are constants that we need to determine.

Inconsistent System

An inconsistent system of linear equations is one that has no solution. In other words, there is no value of $x$ and $y$ that satisfies both equations in the system. To determine the values of $a$ and $b$ that make the system inconsistent, we need to consider the following:

• If the two equations are identical, then the system is inconsistent.
• If the two equations are parallel, then the system is inconsistent.

Case 1: Identical Equations

If the two equations are identical, then we have:

$$3x + 5 = ax + b$$

Subtracting $ax$ from both sides gives:

$$(3-a)x + 5 = b$$

Subtracting $5$ from both sides gives:

$$(3-a)x = b - 5$$

Dividing both sides by $(3-a)$ gives:

$$x = \frac{b - 5}{3-a}$$

However, if $a = 3$, then the equation becomes:

$$x = \frac{b - 5}{0}$$

which is undefined. Therefore, if $a = 3$, then the system is inconsistent.

Case 2: Parallel Equations

If the two equations are parallel, then we have:

$$3x + 5 = ax + b$$

Subtracting $ax$ from both sides gives:

$$(3-a)x + 5 = b$$

Subtracting $5$ from both sides gives:

$$(3-a)x = b - 5$$

Dividing both sides by $(3-a)$ gives:

$$x = \frac{b - 5}{3-a}$$

However, if $a = 3$, then the equation becomes:

$$x = \frac{b - 5}{0}$$

which is undefined. Therefore, if $a = 3$, then the system is inconsistent.

Consistent System

A consistent system of linear equations is one that has a solution. In other words, there is a value of $x$ and $y$ that satisfies both equations in the system. To determine the values of $a$ and $b$ that make the system consistent, we need to consider the following:

• If the two equations are not identical and not parallel, then the system is consistent.

Case 1: Not Identical and Not Parallel

If the two equations are not identical and not parallel, then we have:

$$3x + 5 = ax + b$$

Subtracting $ax$ from both sides gives:

$$(3-a)x + 5 = b$$

Subtracting $5$ from both sides gives:

$$(3-a)x = b - 5$$

Dividing both sides by $(3-a)$ gives:

$$x = \frac{b - 5}{3-a}$$

Therefore, if $a \neq 3$, then the system is consistent.

Conclusion

In conclusion, a system of linear equations is inconsistent if the two equations are identical or parallel. A system of linear equations is consistent if the two equations are not identical and not parallel. Therefore, to determine the values of $a$ and $b$ that make the system inconsistent or consistent, we need to consider the following:

• If $a = 3$, then the system is inconsistent.
• If $a \neq 3$, then the system is consistent.

References

• [1] Linear Algebra and Its Applications, by Gilbert Strang
• [2] Introduction to Linear Algebra, by Jim Hefferon

Further Reading

• Linear Equations and Inequalities
• Systems of Linear Equations
• Linear Algebra

Glossary

• Inconsistent System: A system of linear equations that has no solution.
• Consistent System: A system of linear equations that has a solution.
• Linear Equation: An equation of the form $ax + b = c$.
• Linear Inequality: An inequality of the form $ax + b \geq c$ or $ax + b \leq c$.
• Linear Algebra: A branch of mathematics that deals with the study of linear equations and linear transformations.
  Frequently Asked Questions (FAQs) about Inconsistent and Consistent Systems of Linear Equations =====================================================================================

Q: What is the difference between an inconsistent and a consistent system of linear equations?

A: An inconsistent system of linear equations is one that has no solution, while a consistent system of linear equations is one that has a solution.

Q: How can I determine if a system of linear equations is inconsistent or consistent?

A: To determine if a system of linear equations is inconsistent or consistent, you need to check if the two equations are identical or parallel. If they are, then the system is inconsistent. If they are not, then the system is consistent.

Q: What happens if the two equations in a system of linear equations are identical?

A: If the two equations in a system of linear equations are identical, then the system is inconsistent. This is because there is no value of x and y that satisfies both equations.

Q: What happens if the two equations in a system of linear equations are parallel?

A: If the two equations in a system of linear equations are parallel, then the system is inconsistent. This is because there is no value of x and y that satisfies both equations.

Q: How can I solve a system of linear equations that is consistent?

A: To solve a system of linear equations that is consistent, you need to find the values of x and y that satisfy both equations. This can be done using various methods, such as substitution or elimination.

Q: What is the importance of determining if a system of linear equations is inconsistent or consistent?

A: Determining if a system of linear equations is inconsistent or consistent is important because it helps you to understand the nature of the system and to determine the best method for solving it.

Q: Can a system of linear equations be both inconsistent and consistent at the same time?

A: No, a system of linear equations cannot be both inconsistent and consistent at the same time. A system of linear equations is either inconsistent or consistent, but not both.

Q: How can I represent a system of linear equations graphically?

A: A system of linear equations can be represented graphically by plotting the two equations on a coordinate plane. If the two equations intersect, then the system is consistent. If the two equations do not intersect, then the system is inconsistent.

Q: What is the relationship between a system of linear equations and its solution?

A: A system of linear equations and its solution are related in that the solution is a set of values that satisfy both equations in the system.

Q: Can a system of linear equations have multiple solutions?

A: No, a system of linear equations cannot have multiple solutions. A system of linear equations has either one solution or no solution.

Q: How can I determine if a system of linear equations has a unique solution?

A: To determine if a system of linear equations has a unique solution, you need to check if the two equations are not identical and not parallel. If they are not, then the system has a unique solution.

Q: What is the significance of the concept of inconsistent and consistent systems of linear equations in real-world applications?

A: The concept of inconsistent and consistent systems of linear equations is significant in real-world applications because it helps to model and solve problems in various fields, such as physics, engineering, economics, and computer science.

Q: Can a system of linear equations be used to model real-world problems that involve multiple variables?

A: Yes, a system of linear equations can be used to model real-world problems that involve multiple variables. This is because a system of linear equations can be used to represent a set of linear equations that involve multiple variables.

Q: How can I use a system of linear equations to solve a problem that involves multiple variables?

A: To use a system of linear equations to solve a problem that involves multiple variables, you need to first represent the problem as a system of linear equations. Then, you can use various methods, such as substitution or elimination, to solve the system and find the values of the variables.

Q: What are some common applications of systems of linear equations in real-world problems?

A: Some common applications of systems of linear equations in real-world problems include:

• Modeling population growth and decline
• Solving problems in physics and engineering
• Analyzing data in economics and finance
• Solving problems in computer science and programming
• Modeling supply and demand in economics

Q: Can a system of linear equations be used to solve problems that involve non-linear equations?

A: No, a system of linear equations cannot be used to solve problems that involve non-linear equations. A system of linear equations is used to solve problems that involve linear equations, while non-linear equations require different methods and techniques to solve.

Q: How can I determine if a system of linear equations is linear or non-linear?

A: To determine if a system of linear equations is linear or non-linear, you need to check if the equations are linear or non-linear. If the equations are linear, then the system is linear. If the equations are non-linear, then the system is non-linear.