How Many Solutions Does This Linear System Have?$ \begin{array}{l} y = 2x - 5 \ -8x - 4y = -20 \end{array} }$A. One Solution { (-2.5, 0)$ $ B. One Solution: { (2.5, 0)$}$ C. No Solution D. Infinite Number Of Solutions

by ADMIN 222 views

Introduction

Linear systems are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and economics. A linear system consists of two or more linear equations that involve variables raised to the power of one. In this article, we will focus on a specific linear system and explore the different possibilities for the number of solutions it may have.

The Linear System

The given linear system is:

y=2xβˆ’5βˆ’8xβˆ’4y=βˆ’20\begin{array}{l} y = 2x - 5 \\ -8x - 4y = -20 \end{array}

To solve this system, we can use the method of substitution or elimination. Let's use the substitution method to find the solution.

Substitution Method

We can start by substituting the expression for y from the first equation into the second equation:

βˆ’8xβˆ’4(2xβˆ’5)=βˆ’20-8x - 4(2x - 5) = -20

Expanding and simplifying the equation, we get:

βˆ’8xβˆ’8x+20=βˆ’20-8x - 8x + 20 = -20

Combine like terms:

βˆ’16x+20=βˆ’20-16x + 20 = -20

Subtract 20 from both sides:

βˆ’16x=βˆ’40-16x = -40

Divide both sides by -16:

x=2.5x = 2.5

Now that we have found the value of x, we can substitute it back into one of the original equations to find the value of y. Let's use the first equation:

y=2xβˆ’5y = 2x - 5

Substitute x = 2.5:

y=2(2.5)βˆ’5y = 2(2.5) - 5

Simplify:

y=5βˆ’5y = 5 - 5

y=0y = 0

Therefore, the solution to the linear system is (2.5, 0).

Understanding the Number of Solutions

Now that we have found the solution to the linear system, let's explore the different possibilities for the number of solutions.

One Solution

In this case, we have found a unique solution (2.5, 0) that satisfies both equations. This means that the linear system has exactly one solution.

No Solution

If the linear system has no solution, it means that the equations are inconsistent, and there is no value of x and y that can satisfy both equations.

Infinite Number of Solutions

If the linear system has an infinite number of solutions, it means that the equations are dependent, and there are infinitely many values of x and y that can satisfy both equations.

Conclusion

In conclusion, the linear system has exactly one solution, which is (2.5, 0). This means that the equations are consistent, and there is a unique value of x and y that can satisfy both equations.

Final Answer

The final answer is:

  • A. One solution: (2.5, 0)

This is the correct answer, as we have found a unique solution to the linear system.

Discussion

The discussion category for this problem is mathematics. This is because the problem involves solving a linear system, which is a fundamental concept in mathematics.

Related Topics

Some related topics to this problem include:

  • Linear Equations: Linear equations are equations that involve variables raised to the power of one. They can be used to model a wide range of real-world problems.
  • Systems of Equations: Systems of equations are sets of two or more linear equations that involve variables raised to the power of one. They can be used to model a wide range of real-world problems.
  • Substitution Method: The substitution method is a technique used to solve systems of equations. It involves substituting the expression for one variable into the other equation.
  • Elimination Method: The elimination method is a technique used to solve systems of equations. It involves adding or subtracting the equations to eliminate one of the variables.

Key Concepts

Some key concepts to this problem include:

  • Linear System: A linear system is a set of two or more linear equations that involve variables raised to the power of one.
  • Solution: A solution to a linear system is a value of x and y that can satisfy both equations.
  • Consistent Equations: Consistent equations are equations that have a solution. Inconsistent equations are equations that have no solution.
  • Dependent Equations: Dependent equations are equations that have an infinite number of solutions.

Introduction

In our previous article, we explored the concept of linear systems and how to solve them. We also discussed the different possibilities for the number of solutions a linear system may have. In this article, we will answer some frequently asked questions about solving linear systems.

Q: What is a linear system?

A: A linear system is a set of two or more linear equations that involve variables raised to the power of one. It can be used to model a wide range of real-world problems.

Q: How do I solve a linear system?

A: There are several methods to solve a linear system, including the substitution method and the elimination method. The substitution method involves substituting the expression for one variable into the other equation, while the elimination method involves adding or subtracting the equations to eliminate one of the variables.

Q: What is the difference between a consistent and inconsistent equation?

A: A consistent equation is an equation that has a solution, while an inconsistent equation is an equation that has no solution. In a consistent equation, the equations are either dependent or independent, while in an inconsistent equation, the equations are contradictory.

Q: What is the difference between a dependent and independent equation?

A: A dependent equation is an equation that has an infinite number of solutions, while an independent equation is an equation that has a unique solution. In a dependent equation, the equations are equivalent, while in an independent equation, the equations are distinct.

Q: How do I determine if a linear system has one solution, no solution, or an infinite number of solutions?

A: To determine the number of solutions a linear system has, you can use the following steps:

  • Check if the equations are consistent or inconsistent.
  • Check if the equations are dependent or independent.
  • If the equations are consistent and independent, the linear system has one solution.
  • If the equations are consistent and dependent, the linear system has an infinite number of solutions.
  • If the equations are inconsistent, the linear system has no solution.

Q: What are some common mistakes to avoid when solving linear systems?

A: Some common mistakes to avoid when solving linear systems include:

  • Not checking if the equations are consistent or inconsistent.
  • Not checking if the equations are dependent or independent.
  • Not using the correct method to solve the linear system.
  • Not checking for extraneous solutions.

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you can substitute the solution back into the original equations and check if it satisfies both equations. If the solution does not satisfy both equations, it is an extraneous solution.

Q: What are some real-world applications of linear systems?

A: Linear systems have many real-world applications, including:

  • Physics: Linear systems are used to model the motion of objects under the influence of forces.
  • Engineering: Linear systems are used to model the behavior of electrical circuits and mechanical systems.
  • Economics: Linear systems are used to model the behavior of economic systems and make predictions about future trends.
  • Computer Science: Linear systems are used to model the behavior of computer networks and make predictions about future trends.

Conclusion

In conclusion, solving linear systems is a crucial concept in mathematics and has many real-world applications. By understanding the different possibilities for the number of solutions a linear system may have, you can use the correct method to solve the linear system and make predictions about future trends.

Final Answer

The final answer is:

  • A linear system has one solution, no solution, or an infinite number of solutions, depending on whether the equations are consistent and independent, consistent and dependent, or inconsistent.

This is the correct answer, as it summarizes the different possibilities for the number of solutions a linear system may have.

Discussion

The discussion category for this problem is mathematics. This is because the problem involves solving linear systems, which is a fundamental concept in mathematics.

Related Topics

Some related topics to this problem include:

  • Linear Equations: Linear equations are equations that involve variables raised to the power of one. They can be used to model a wide range of real-world problems.
  • Systems of Equations: Systems of equations are sets of two or more linear equations that involve variables raised to the power of one. They can be used to model a wide range of real-world problems.
  • Substitution Method: The substitution method is a technique used to solve systems of equations. It involves substituting the expression for one variable into the other equation.
  • Elimination Method: The elimination method is a technique used to solve systems of equations. It involves adding or subtracting the equations to eliminate one of the variables.

Key Concepts

Some key concepts to this problem include:

  • Linear System: A linear system is a set of two or more linear equations that involve variables raised to the power of one.
  • Solution: A solution to a linear system is a value of x and y that can satisfy both equations.
  • Consistent Equations: Consistent equations are equations that have a solution. Inconsistent equations are equations that have no solution.
  • Dependent Equations: Dependent equations are equations that have an infinite number of solutions.