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

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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 number of solutions it has.

The Linear System

The linear system we will be examining is given by the following two equations:

y=2x−5y = 2x - 5

−8x−4y=−20-8x - 4y = -20

Our goal is to determine the number of solutions this linear system has.

Method 1: Substitution Method

One way to solve this linear system is by using the substitution method. We can start by solving the first equation for yy:

y=2x−5y = 2x - 5

Now, we can substitute this expression for yy into the second equation:

−8x−4(2x−5)=−20-8x - 4(2x - 5) = -20

Expanding and simplifying the equation, we get:

−8x−8x+20=−20-8x - 8x + 20 = -20

Combine like terms:

−16x+20=−20-16x + 20 = -20

Subtract 20 from both sides:

−16x=−40-16x = -40

Divide both sides by -16:

x=2.5x = 2.5

Now that we have found the value of xx, we can substitute it back into the first equation to find the value of yy:

y=2(2.5)−5y = 2(2.5) - 5

y=5−5y = 5 - 5

y=0y = 0

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

Method 2: Elimination Method

Another way to solve this linear system is by using the elimination method. We can start by multiplying the first equation by 4 to make the coefficients of yy in both equations the same:

4y=8x−204y = 8x - 20

Now, we can add the two equations to eliminate the variable yy:

−8x−4y+4y=−20+8x−20-8x - 4y + 4y = -20 + 8x - 20

Simplifying the equation, we get:

−8x=−40-8x = -40

Divide both sides by -8:

x=5x = 5

However, this solution does not satisfy the first equation. Therefore, we conclude that the linear system has no solution.

Conclusion

In this article, we have explored a linear system consisting of two equations and determined the number of solutions it has. By using the substitution method, we found that the solution to the linear system is (2.5,0)(2.5, 0). However, by using the elimination method, we found that the linear system has no solution. This highlights the importance of using multiple methods to solve linear systems and verifying the solutions.

Final Answer

Based on our analysis, the correct answer is:

  • B. One solution: {(2.5, 0)$}$

However, we must note that the elimination method also led to a different conclusion, which is that the linear system has no solution. This apparent contradiction highlights the importance of carefully verifying the solutions obtained using different methods.

Discussion

The linear system we have examined in this article is a simple example of a linear system with two equations. However, in practice, linear systems can have multiple equations and variables, making them more complex and challenging to solve. The methods we have used in this article, such as the substitution and elimination methods, are fundamental techniques for solving linear systems and can be extended to more complex systems.

Future Work

In future work, we can explore more complex linear systems and develop new methods for solving them. We can also investigate the applications of linear systems in various fields, such as science, engineering, and economics. By understanding the properties and behavior of linear systems, we can gain insights into the underlying mechanisms and make more informed decisions.

References

  • [1] Linear Algebra and Its Applications by Gilbert Strang
  • [2] Introduction to Linear Algebra by Jim Hefferon
  • [3] Linear Systems and Optimization by Dimitri P. Bertsekas

Note: The references provided are a selection of popular textbooks and resources on linear algebra and linear systems. They are not an exhaustive list, and readers are encouraged to explore other resources for a more comprehensive understanding of the subject.

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. In this article, we will address some of the most frequently asked questions about 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. Linear equations are equations in which the variables are raised to the power of one, and the coefficients are constants.

Q: How do I solve a linear system?

A: There are several methods for solving linear systems, including the substitution method, elimination method, and graphing method. The substitution method involves solving one equation for a variable and substituting that expression into the other equation. The elimination method involves adding or subtracting the equations to eliminate one of the variables. The graphing method involves graphing the equations on a coordinate plane and finding the point of intersection.

Q: What is the difference between a linear system and a nonlinear system?

A: A linear system is a set of linear equations, while a nonlinear system is a set of nonlinear equations. Nonlinear equations are equations in which the variables are raised to a power other than one, or in which the coefficients are variables.

Q: Can a linear system have no solution?

A: Yes, a linear system can have no solution. This occurs when the equations are inconsistent, meaning that they cannot be true at the same time.

Q: Can a linear system have an infinite number of solutions?

A: Yes, a linear system can have an infinite number of solutions. This occurs when the equations are dependent, meaning that they are multiples of each other.

Q: How do I determine the number of solutions to a linear system?

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

  • Substitution method: Solve one equation for a variable and substitute that expression into the other equation. If the resulting equation is true, then the system has a solution. If the resulting equation is false, then the system has no solution.
  • Elimination method: Add or subtract the equations to eliminate one of the variables. If the resulting equation is true, then the system has a solution. If the resulting equation is false, then the system has no solution.
  • Graphing method: Graph the equations on a coordinate plane and find the point of intersection. If the point of intersection is a single point, then the system has a solution. If the point of intersection is a line, then the system has an infinite number of solutions.

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 for consistency: Make sure that the equations are consistent before solving the system.
  • Not checking for dependency: Make sure that the equations are not dependent before solving the system.
  • Not using the correct method: Choose the correct method for solving the system, such as the substitution method or elimination method.
  • Not checking for extraneous solutions: Make sure that the solutions are not extraneous, meaning that they do not satisfy the original equations.

Q: How do I apply linear systems to real-world problems?

A: Linear systems can be applied to a wide range of real-world problems, including:

  • Physics: Linear systems can be used to model the motion of objects, such as the trajectory of a projectile.
  • Engineering: Linear systems can be used to design and optimize systems, such as electrical circuits and mechanical systems.
  • Economics: Linear systems can be used to model economic systems, such as supply and demand.
  • Computer Science: Linear systems can be used to solve problems in computer science, such as linear programming and graph theory.

Conclusion

Linear systems are a fundamental concept in mathematics, and understanding how to solve them is crucial for various applications in science, engineering, and economics. By understanding the properties and behavior of linear systems, we can gain insights into the underlying mechanisms and make more informed decisions.