Solve The Following System Of Equations:${ \begin{array}{l} 2x - 2y = 4 \ -2x + 2y = -7 \end{array} }$

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Introduction

In mathematics, a system of linear equations is a set of two or more linear equations that involve the same set of 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 focus on solving a system of two linear equations with two variables.

The System of Equations

The system of equations we will be solving is:

{ \begin{array}{l} 2x - 2y = 4 \\ -2x + 2y = -7 \end{array} \}

Step 1: Write Down the System of Equations

The first step in solving a system of linear equations is to write down the system of equations. In this case, we have two equations:

  1. 2x−2y=42x - 2y = 4
  2. −2x+2y=−7-2x + 2y = -7

Step 2: Add the Two Equations

To eliminate one of the variables, we can add the two equations together. This will eliminate the variable yy.

{ \begin{array}{l} (2x - 2y) + (-2x + 2y) = 4 + (-7) \\ 0 = -3 \end{array} \}

However, this is not a valid solution, as the equation 0=−30 = -3 is not true. This means that the two equations are inconsistent, and there is no solution to the system.

Step 3: Check for Inconsistency

Since the two equations are inconsistent, we can conclude that there is no solution to the system. This is because the two equations are contradictory, and it is not possible for both equations to be true at the same time.

Conclusion

In this article, we solved a system of two linear equations with two variables. We found that the two equations are inconsistent, and there is no solution to the system. This is because the two equations are contradictory, and it is not possible for both equations to be true at the same time.

Tips and Tricks

  • When solving a system of linear equations, it is essential to check for inconsistency before trying to find a solution.
  • If the two equations are inconsistent, there is no solution to the system.
  • Inconsistent equations can be identified by adding the two equations together and checking if the result is a true statement.

Real-World Applications

Solving systems of linear equations has many real-world applications, including:

  • Physics and Engineering: Systems of linear equations are used to model real-world problems, such as the motion of objects and the behavior of electrical circuits.
  • Economics: Systems of linear equations are used to model economic systems, such as supply and demand curves.
  • Computer Science: Systems of linear equations are used in computer science to solve problems, such as linear programming and graph theory.

Common Mistakes

  • Not checking for inconsistency: Failing to check for inconsistency can lead to incorrect solutions.
  • Not using the correct method: Using the wrong method to solve a system of linear equations can lead to incorrect solutions.
  • Not checking the work: Failing to check the work can lead to incorrect solutions.

Conclusion

Introduction

In our previous article, we discussed how to solve a system of linear equations. However, we know that there are many questions that readers may have about solving systems of linear equations. In this article, we will answer some of the most frequently asked questions about solving systems of linear equations.

Q: What is a system of linear equations?

A: A system of linear equations is a set of two or more linear equations that involve the same set of variables. For example:

{ \begin{array}{l} 2x - 2y = 4 \\ -2x + 2y = -7 \end{array} \}

Q: How do I know if a system of linear equations has a solution?

A: To determine if a system of linear equations has a solution, you need to check if the two equations are consistent. If the two equations are consistent, then the system has a solution. If the two equations are inconsistent, then the system does not have a solution.

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

A: A consistent system of linear equations is one where the two equations are not contradictory. For example:

{ \begin{array}{l} 2x - 2y = 4 \\ -2x + 2y = 2 \end{array} \}

This system is consistent because the two equations are not contradictory.

On the other hand, an inconsistent system of linear equations is one where the two equations are contradictory. For example:

{ \begin{array}{l} 2x - 2y = 4 \\ -2x + 2y = -7 \end{array} \}

This system is inconsistent because the two equations are contradictory.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you need to follow these steps:

  1. Write down the system of linear equations.
  2. Check if the two equations are consistent.
  3. If the two equations are consistent, then you can use the method of substitution or elimination to solve the system.
  4. If the two equations are inconsistent, then the system does not have a solution.

Q: What is the method of substitution?

A: The method of substitution is a technique used to solve a system of linear equations. It involves solving one equation for one variable and then substituting that expression into the other equation.

For example, consider the system of linear equations:

{ \begin{array}{l} 2x - 2y = 4 \\ -2x + 2y = -7 \end{array} \}

To solve this system using the method of substitution, we can solve the first equation for x:

x = (4 + 2y) / 2

Then, we can substitute this expression into the second equation:

-2((4 + 2y) / 2) + 2y = -7

Simplifying this equation, we get:

-4 - 2y + 2y = -7

This equation is inconsistent, so the system does not have a solution.

Q: What is the method of elimination?

A: The method of elimination is a technique used to solve a system of linear equations. It involves adding or subtracting the two equations to eliminate one of the variables.

For example, consider the system of linear equations:

{ \begin{array}{l} 2x - 2y = 4 \\ -2x + 2y = -7 \end{array} \}

To solve this system using the method of elimination, we can add the two equations together:

(2x - 2y) + (-2x + 2y) = 4 + (-7)

This equation is inconsistent, so the system does not have a solution.

Conclusion

In this article, we answered some of the most frequently asked questions about solving systems of linear equations. We discussed the difference between a consistent and an inconsistent system of linear equations, and we explained how to solve a system of linear equations using the method of substitution and elimination. We also provided examples of how to use these techniques to solve systems of linear equations.