Solve The System Using Elimination.$\[ \begin{array}{l} 6x - 6y = -24 \\ 7x + 7y = -14 \end{array} \\]The Solution Is $\square$. (Simplify Your Answer. Type An Ordered Pair.)

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Introduction

In mathematics, solving systems of linear equations is a fundamental concept that involves finding the values of variables that satisfy multiple equations simultaneously. There are several methods to solve systems of linear equations, including substitution, elimination, and graphing. In this article, we will focus on the elimination method, which involves adding or subtracting equations to eliminate variables and solve for the remaining variables.

What is the Elimination Method?

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate variables. This method involves creating a new equation by adding or subtracting the two original equations, which results in a single equation with one variable. The value of the variable can then be determined by solving the resulting equation.

Step-by-Step Guide to Solving Systems using Elimination

To solve a system of linear equations using the elimination method, follow these steps:

  1. Write down the two equations: Write down the two linear equations that make up the system.
  2. Identify the coefficients: Identify the coefficients of the variables in both equations.
  3. Multiply the equations: Multiply both equations by necessary multiples such that the coefficients of the variables to be eliminated are the same.
  4. Add or subtract the equations: Add or subtract the two equations to eliminate the variable.
  5. Solve for the remaining variable: Solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.
  6. Find the value of the other variable: Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Example: Solving the System using Elimination

Let's consider the following system of linear equations:

{ \begin{array}{l} 6x - 6y = -24 \\ 7x + 7y = -14 \end{array} \}

To solve this system using the elimination method, follow these steps:

  1. Write down the two equations: Write down the two linear equations that make up the system.

{ \begin{array}{l} 6x - 6y = -24 \\ 7x + 7y = -14 \end{array} \}

  1. Identify the coefficients: Identify the coefficients of the variables in both equations.

In the first equation, the coefficient of x is 6 and the coefficient of y is -6. In the second equation, the coefficient of x is 7 and the coefficient of y is 7.

  1. Multiply the equations: Multiply both equations by necessary multiples such that the coefficients of the variables to be eliminated are the same.

To eliminate the variable y, multiply the first equation by 1 and the second equation by 1.

{ \begin{array}{l} 6x - 6y = -24 \\ 7x + 7y = -14 \end{array} \}

  1. Add or subtract the equations: Add or subtract the two equations to eliminate the variable.

Add the two equations to eliminate the variable y.

{ \begin{array}{l} (6x - 6y) + (7x + 7y) = -24 + (-14) \\ 13x = -38 \end{array} \}

  1. Solve for the remaining variable: Solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.

Divide both sides of the equation by 13 to solve for x.

{ \begin{array}{l} 13x = -38 \\ x = -38/13 \\ x = -2.923 \end{array} \}

  1. Find the value of the other variable: Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Substitute the value of x into the first equation to find the value of y.

{ \begin{array}{l} 6x - 6y = -24 \\ 6(-2.923) - 6y = -24 \\ -17.538 - 6y = -24 \\ -6y = -6.462 \\ y = 1.077 \end{array} \}

Therefore, the solution to the system is (-2.923, 1.077).

Conclusion

In this article, we have discussed the elimination method for solving systems of linear equations. We have provided a step-by-step guide to solving systems using the elimination method and have used an example to illustrate the process. The elimination method is a powerful tool for solving systems of linear equations and is widely used in mathematics and other fields.

Frequently Asked Questions

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate variables.

Q: How do I use the elimination method to solve a system of linear equations?

A: To use the elimination method, follow these steps: write down the two equations, identify the coefficients, multiply the equations, add or subtract the equations, solve for the remaining variable, and find the value of the other variable.

Q: What are the advantages of the elimination method?

A: The elimination method has several advantages, including the ability to solve systems of linear equations quickly and easily, and the ability to find the values of variables that satisfy multiple equations simultaneously.

Q: What are the disadvantages of the elimination method?

A: The elimination method has several disadvantages, including the requirement that the coefficients of the variables to be eliminated are the same, and the potential for errors when multiplying and adding or subtracting equations.

References

  • [1] "Linear Algebra and Its Applications" by Gilbert Strang
  • [2] "Introduction to Linear Algebra" by Jim Hefferon
  • [3] "Solving Systems of Linear Equations" by Math Open Reference

Additional Resources

  • [1] Khan Academy: Solving Systems of Linear Equations
  • [2] MIT OpenCourseWare: Linear Algebra
  • [3] Wolfram Alpha: Solving Systems of Linear Equations
    Frequently Asked Questions about Solving Systems of Linear Equations using Elimination =====================================================================================

Q: What is the elimination method?

A: The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate variables. This method involves creating a new equation by adding or subtracting the two original equations, which results in a single equation with one variable.

Q: How do I use the elimination method to solve a system of linear equations?

A: To use the elimination method, follow these steps:

  1. Write down the two equations: Write down the two linear equations that make up the system.
  2. Identify the coefficients: Identify the coefficients of the variables in both equations.
  3. Multiply the equations: Multiply both equations by necessary multiples such that the coefficients of the variables to be eliminated are the same.
  4. Add or subtract the equations: Add or subtract the two equations to eliminate the variable.
  5. Solve for the remaining variable: Solve for the remaining variable by dividing both sides of the equation by the coefficient of the variable.
  6. Find the value of the other variable: Substitute the value of the remaining variable into one of the original equations to find the value of the other variable.

Q: What are the advantages of the elimination method?

A: The elimination method has several advantages, including:

  • Quick and easy solution: The elimination method can solve systems of linear equations quickly and easily.
  • Ability to find multiple solutions: The elimination method can find the values of variables that satisfy multiple equations simultaneously.
  • Simple to understand: The elimination method is a simple and intuitive method to understand.

Q: What are the disadvantages of the elimination method?

A: The elimination method has several disadvantages, including:

  • Requires careful calculation: The elimination method requires careful calculation to ensure that the coefficients of the variables to be eliminated are the same.
  • Potential for errors: The elimination method can be prone to errors if the calculations are not done correctly.
  • Limited applicability: The elimination method is not suitable for all types of systems of linear equations.

Q: When should I use the elimination method?

A: You should use the elimination method when:

  • The system of linear equations has two equations: The elimination method is most effective when the system of linear equations has two equations.
  • The coefficients of the variables to be eliminated are the same: The elimination method requires that the coefficients of the variables to be eliminated are the same.
  • You want a quick and easy solution: The elimination method is a quick and easy method to solve systems of linear equations.

Q: When should I not use the elimination method?

A: You should not use the elimination method when:

  • The system of linear equations has more than two equations: The elimination method is not suitable for systems of linear equations with more than two equations.
  • The coefficients of the variables to be eliminated are not the same: The elimination method requires that the coefficients of the variables to be eliminated are the same.
  • You want to find the values of all variables: The elimination method is not suitable for finding the values of all variables in a system of linear equations.

Q: How do I choose between the elimination method and the substitution method?

A: You should choose between the elimination method and the substitution method based on the following factors:

  • The complexity of the system of linear equations: If the system of linear equations is simple, the substitution method may be more effective. If the system of linear equations is complex, the elimination method may be more effective.
  • The number of equations: If the system of linear equations has two equations, the elimination method may be more effective. If the system of linear equations has more than two equations, the substitution method may be more effective.
  • Your personal preference: Ultimately, the choice between the elimination method and the substitution method depends on your personal preference.

Q: Can I use the elimination method to solve systems of linear equations with fractions?

A: Yes, you can use the elimination method to solve systems of linear equations with fractions. However, you will need to follow the same steps as before, but with the added complexity of fractions.

Q: Can I use the elimination method to solve systems of linear equations with decimals?

A: Yes, you can use the elimination method to solve systems of linear equations with decimals. However, you will need to follow the same steps as before, but with the added complexity of decimals.

Q: Can I use the elimination method to solve systems of linear equations with negative numbers?

A: Yes, you can use the elimination method to solve systems of linear equations with negative numbers. However, you will need to follow the same steps as before, but with the added complexity of negative numbers.

Q: Can I use the elimination method to solve systems of linear equations with variables on both sides?

A: Yes, you can use the elimination method to solve systems of linear equations with variables on both sides. However, you will need to follow the same steps as before, but with the added complexity of variables on both sides.

Q: Can I use the elimination method to solve systems of linear equations with multiple variables?

A: Yes, you can use the elimination method to solve systems of linear equations with multiple variables. However, you will need to follow the same steps as before, but with the added complexity of multiple variables.

Q: Can I use the elimination method to solve systems of linear equations with non-linear equations?

A: No, you cannot use the elimination method to solve systems of linear equations with non-linear equations. The elimination method is only suitable for solving systems of linear equations.

Q: Can I use the elimination method to solve systems of linear equations with complex numbers?

A: No, you cannot use the elimination method to solve systems of linear equations with complex numbers. The elimination method is only suitable for solving systems of linear equations with real numbers.

Q: Can I use the elimination method to solve systems of linear equations with matrices?

A: No, you cannot use the elimination method to solve systems of linear equations with matrices. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with vectors?

A: No, you cannot use the elimination method to solve systems of linear equations with vectors. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with functions?

A: No, you cannot use the elimination method to solve systems of linear equations with functions. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with polynomials?

A: No, you cannot use the elimination method to solve systems of linear equations with polynomials. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with rational expressions?

A: No, you cannot use the elimination method to solve systems of linear equations with rational expressions. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with trigonometric functions?

A: No, you cannot use the elimination method to solve systems of linear equations with trigonometric functions. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with exponential functions?

A: No, you cannot use the elimination method to solve systems of linear equations with exponential functions. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with logarithmic functions?

A: No, you cannot use the elimination method to solve systems of linear equations with logarithmic functions. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with absolute value functions?

A: No, you cannot use the elimination method to solve systems of linear equations with absolute value functions. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with piecewise functions?

A: No, you cannot use the elimination method to solve systems of linear equations with piecewise functions. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with parametric equations?

A: No, you cannot use the elimination method to solve systems of linear equations with parametric equations. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with polar equations?

A: No, you cannot use the elimination method to solve systems of linear equations with polar equations. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

Q: Can I use the elimination method to solve systems of linear equations with parametric equations with multiple variables?

A: No, you cannot use the elimination method to solve systems of linear equations with parametric equations with multiple variables. The elimination method is only suitable for solving systems of linear equations with scalar coefficients.

**Q: Can I use the elimination method to solve systems of linear equations