Without Graphing, Classify The Following System As Independent, Dependent, Or Inconsistent.$\[ \begin{cases} -x - 2y = -7 \\ 5x - Y = 2 \end{cases} \\]A. DependentB. IndependentC. Inconsistent
Introduction
In mathematics, a system of linear equations is a set of two or more linear equations that involve the same variables. Solving systems of linear equations is a fundamental concept in algebra and is used to find the solution to a system of equations. In this article, we will discuss how to classify a system of linear equations as independent, dependent, or inconsistent without graphing.
What are Independent, Dependent, and Inconsistent Systems?
Before we dive into the classification of systems, let's define what independent, dependent, and inconsistent systems are.
- Independent System: An independent system is a system of linear equations that has a unique solution. In other words, the system has a single solution that satisfies both equations.
- Dependent System: A dependent system is a system of linear equations that has an infinite number of solutions. This occurs when one equation is a multiple of the other equation.
- Inconsistent System: An inconsistent system is a system of linear equations that has no solution. This occurs when the equations are contradictory, meaning that they cannot be true at the same time.
Classifying Systems Without Graphing
To classify a system of linear equations without graphing, we can use the method of substitution or elimination. Let's use the given system of equations as an example:
{ \begin{cases} -x - 2y = -7 \\ 5x - y = 2 \end{cases} \}
Step 1: Multiply the Equations by Necessary Multiples
To eliminate one of the variables, we need to multiply the equations by necessary multiples such that the coefficients of the variable to be eliminated are the same.
Let's multiply the first equation by 5 and the second equation by 1:
{ \begin{cases} -5x - 10y = -35 \\ 5x - y = 2 \end{cases} \}
Step 2: Add or Subtract the Equations
Now, let's add the two equations to eliminate the variable x:
{ -10y - y = -33 \}
Simplifying the equation, we get:
{ -11y = -33 \}
Step 3: Solve for the Variable
Now, let's solve for the variable y:
{ y = \frac{-33}{-11} \}
Simplifying the equation, we get:
{ y = 3 \}
Step 4: Substitute the Value of the Variable
Now, let's substitute the value of y into one of the original equations to solve for the variable x. Let's use the second equation:
{ 5x - y = 2 \}
Substituting y = 3, we get:
{ 5x - 3 = 2 \}
Simplifying the equation, we get:
{ 5x = 5 \}
Solving for x, we get:
{ x = 1 \}
Conclusion
In this article, we discussed how to classify a system of linear equations as independent, dependent, or inconsistent without graphing. We used the method of substitution and elimination to solve the system of equations and found that the system has a unique solution, making it an independent system.
Key Takeaways
- A system of linear equations can be classified as independent, dependent, or inconsistent.
- An independent system has a unique solution.
- A dependent system has an infinite number of solutions.
- An inconsistent system has no solution.
- The method of substitution and elimination can be used to solve systems of linear equations.
Practice Problems
Try solving the following systems of linear equations and classify them as independent, dependent, or inconsistent:
{ \begin{cases} 2x + 3y = 7 \\ 4x + 6y = 14 \end{cases} \}
{ \begin{cases} x - 2y = 3 \\ 2x - 4y = 6 \end{cases} \}
{ \begin{cases} x + 2y = 5 \\ 3x + 6y = 15 \end{cases} \}
Answer Key
- Dependent
- Inconsistent
- Independent
Frequently Asked Questions (FAQs) About Classifying Systems of Linear Equations =====================================================================================
Q: What is the difference between an independent and a dependent system of linear equations?
A: An independent system of linear equations has a unique solution, while a dependent system has an infinite number of solutions. In other words, an independent system has a single solution that satisfies both equations, while a dependent system has multiple solutions that satisfy both equations.
Q: How can I determine if a system of linear equations is independent or dependent?
A: To determine if a system of linear equations is independent or dependent, you can use the method of substitution or elimination. If the equations are not multiples of each other, then the system is independent. If the equations are multiples of each other, then the system is dependent.
Q: What is an inconsistent system of linear equations?
A: An inconsistent system of linear equations is a system that has no solution. This occurs when the equations are contradictory, meaning that they cannot be true at the same time.
Q: How can I determine if a system of linear equations is inconsistent?
A: To determine if a system of linear equations is inconsistent, you can use the method of substitution or elimination. If the equations are contradictory, then the system is inconsistent.
Q: What is the difference between a dependent and an inconsistent system of linear equations?
A: A dependent system of linear equations has an infinite number of solutions, while an inconsistent system has no solution. In other words, a dependent system has multiple solutions that satisfy both equations, while an inconsistent system has no solution that satisfies both equations.
Q: Can a system of linear equations be both dependent and inconsistent?
A: No, a system of linear equations cannot be both dependent and inconsistent. A system can be either dependent or inconsistent, but not both.
Q: How can I classify a system of linear equations as independent, dependent, or inconsistent without graphing?
A: To classify a system of linear equations as independent, dependent, or inconsistent without graphing, you can use the method of substitution or elimination. This involves multiplying the equations by necessary multiples, adding or subtracting the equations, and solving for the variables.
Q: What are some common mistakes to avoid when classifying systems of linear equations?
A: Some common mistakes to avoid when classifying systems of linear equations include:
- Not checking if the equations are multiples of each other
- Not solving for the variables correctly
- Not checking if the equations are contradictory
- Not using the correct method of substitution or elimination
Q: How can I practice classifying systems of linear equations?
A: You can practice classifying systems of linear equations by trying out different systems and using the method of substitution or elimination to classify them. You can also try solving systems of linear equations and then classifying them as independent, dependent, or inconsistent.
Q: What are some real-world applications of classifying systems of linear equations?
A: Classifying systems of linear equations has many real-world applications, including:
- Solving problems in physics and engineering
- Modeling population growth and decline
- Analyzing financial data
- Solving problems in computer science and cryptography
Conclusion
In this article, we discussed frequently asked questions about classifying systems of linear equations. We covered topics such as the difference between independent and dependent systems, how to determine if a system is independent or dependent, and how to classify a system as inconsistent. We also discussed common mistakes to avoid and provided some real-world applications of classifying systems of linear equations.