Which Value, When Placed In The Box, Would Result In A System Of Equations With Infinitely Many Solutions?${ \begin{array}{l} y = 2x - 5 \ 2y - 4x = \text{[box]} \end{array} }$A. { -10$}$B. { -5$}$C. 5D. 10

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Introduction

When dealing with systems of linear equations, we often encounter scenarios where the system has either a unique solution, no solution, or infinitely many solutions. In this article, we will focus on the latter case, where the system of equations has infinitely many solutions. We will explore the conditions under which this occurs and provide a step-by-step guide on how to identify such systems.

What are Systems of Equations?

A system of equations is a set of two or more equations that contain the same variables. These equations are often linear, meaning they are in the form of a straight line. The goal of solving a system of equations is to find the values of the variables that satisfy all the equations simultaneously.

Types of Solutions

When solving a system of equations, we can encounter three types of solutions:

  1. Unique Solution: A system of equations has a unique solution when there is only one set of values that satisfies all the equations.
  2. No Solution: A system of equations has no solution when there is no set of values that satisfies all the equations.
  3. Infinitely Many Solutions: A system of equations has infinitely many solutions when there are an infinite number of sets of values that satisfy all the equations.

Conditions for Infinitely Many Solutions

For a system of equations to have infinitely many solutions, the following conditions must be met:

  1. The equations must be linear: The equations must be in the form of a straight line, i.e., they must be linear.
  2. The equations must be dependent: The equations must be dependent, meaning that one equation can be expressed as a multiple of the other equation.
  3. The equations must have the same slope: The equations must have the same slope, meaning that they must be parallel to each other.

Example: Solving a System of Equations with Infinitely Many Solutions

Let's consider the following system of equations:

{ \begin{array}{l} y = 2x - 5 \\ 2y - 4x = \text{[box]} \end{array} \}

To determine the value that would result in a system of equations with infinitely many solutions, we need to analyze the second equation. If we substitute the expression for y from the first equation into the second equation, we get:

2(2x−5)−4x=[box]{ 2(2x - 5) - 4x = \text{[box]} }

Simplifying the equation, we get:

4x−10−4x=[box]{ 4x - 10 - 4x = \text{[box]} }

This simplifies to:

−10=[box]{ -10 = \text{[box]} }

Therefore, the value that would result in a system of equations with infinitely many solutions is -10.

Conclusion

In conclusion, a system of equations has infinitely many solutions when the equations are linear, dependent, and have the same slope. By analyzing the second equation in the system, we can determine the value that would result in a system of equations with infinitely many solutions. In this article, we have seen how to identify such systems and provide a step-by-step guide on how to solve them.

Final Answer

The final answer is −10\boxed{-10}.

Discussion

Which value, when placed in the box, would result in a system of equations with infinitely many solutions?

A. −10\boxed{-10} B. −5\boxed{-5} C. 5 D. 10

The correct answer is A. −10\boxed{-10}.

Introduction

In our previous article, we explored the concept of systems of equations with infinitely many solutions. We discussed the conditions under which such systems occur and provided a step-by-step guide on how to identify and solve them. In this article, we will answer some frequently asked questions related to systems of equations with infinitely many solutions.

Q&A

Q1: What are the conditions for a system of equations to have infinitely many solutions?

A1: For a system of equations to have infinitely many solutions, the following conditions must be met:

  1. The equations must be linear: The equations must be in the form of a straight line, i.e., they must be linear.
  2. The equations must be dependent: The equations must be dependent, meaning that one equation can be expressed as a multiple of the other equation.
  3. The equations must have the same slope: The equations must have the same slope, meaning that they must be parallel to each other.

Q2: How can I determine if a system of equations has infinitely many solutions?

A2: To determine if a system of equations has infinitely many solutions, you can follow these steps:

  1. Substitute the expression for y from one equation into the other equation: If the resulting equation is true for all values of x, then the system has infinitely many solutions.
  2. Check if the equations are linear and dependent: If the equations are linear and dependent, then the system has infinitely many solutions.
  3. Check if the equations have the same slope: If the equations have the same slope, then the system has infinitely many solutions.

Q3: What is the difference between a system of equations with infinitely many solutions and a system with no solution?

A3: A system of equations with infinitely many solutions is different from a system with no solution in the following ways:

  • Number of solutions: A system with infinitely many solutions has an infinite number of solutions, while a system with no solution has no solutions.
  • Equations: A system with infinitely many solutions has linear and dependent equations, while a system with no solution has linear and independent equations.
  • Slope: A system with infinitely many solutions has equations with the same slope, while a system with no solution has equations with different slopes.

Q4: Can a system of equations have infinitely many solutions if the equations are not linear?

A4: No, a system of equations cannot have infinitely many solutions if the equations are not linear. The conditions for a system of equations to have infinitely many solutions require that the equations be linear and dependent.

Q5: How can I solve a system of equations with infinitely many solutions?

A5: To solve a system of equations with infinitely many solutions, you can follow these steps:

  1. Express one equation in terms of the other equation: If the equations are dependent, then one equation can be expressed as a multiple of the other equation.
  2. Solve for one variable: If the equations are dependent, then you can solve for one variable in terms of the other variable.
  3. Express the solution in terms of a parameter: If the equations are dependent, then you can express the solution in terms of a parameter.

Conclusion

In conclusion, systems of equations with infinitely many solutions are an important concept in mathematics. By understanding the conditions under which such systems occur and how to identify and solve them, you can gain a deeper understanding of the subject. We hope that this Q&A article has been helpful in answering your questions and providing a better understanding of systems of equations with infinitely many solutions.

Final Answer

The final answer is −10\boxed{-10}.

Discussion

Which value, when placed in the box, would result in a system of equations with infinitely many solutions?

A. −10\boxed{-10} B. −5\boxed{-5} C. 5 D. 10

The correct answer is A. −10\boxed{-10}.