Solve The System Using Elimination:${ \begin{array}{c} -3x + 2y = 10 \ 6x + 6y = 0 \ \end{array} }$Provide Your Answer As { (?, \square)$}$.

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Introduction

The elimination method is a technique used to solve systems of linear equations by adding or subtracting equations to eliminate one of the variables. This method is particularly useful when the coefficients of the variables in the equations are relatively simple. In this article, we will use the elimination method to solve a system of two linear equations with two variables.

The System of Linear Equations

The system of linear equations we will be solving is:

{ \begin{array}{c} -3x + 2y = 10 \\ 6x + 6y = 0 \\ \end{array} \}

Step 1: Multiply the Equations by Necessary Multiples

To eliminate one of the variables, we need to make the coefficients of that variable the same in both equations, but with opposite signs. In this case, we can multiply the first equation by 2 and the second equation by 1.

{ \begin{array}{c} -6x + 4y = 20 \\ 6x + 6y = 0 \\ \end{array} \}

Step 2: Add the Equations

Now that the coefficients of x are the same in both equations, but with opposite signs, we can add the equations to eliminate the variable x.

{ \begin{array}{c} -6x + 4y = 20 \\ 6x + 6y = 0 \\ \hline 10y = 20 \\ \end{array} \}

Step 3: Solve for y

Now that we have eliminated the variable x, we can solve for y by dividing both sides of the equation by 10.

{ y = \frac{20}{10} \\ y = 2 \\ \end{array} \}

Step 4: Substitute y into One of the Original Equations

Now that we have found the value of y, we can substitute it into one of the original equations to solve for x. We will use the first equation.

{ -3x + 2y = 10 \\ -3x + 2(2) = 10 \\ -3x + 4 = 10 \\ -3x = 6 \\ x = -2 \\ \end{array} \}

Conclusion

We have successfully solved the system of linear equations using the elimination method. The solution to the system is (x, y) = (-2, 2).

Example Use Case

The elimination method can be used to solve systems of linear equations in a variety of real-world applications, such as:

  • Finding the intersection point of two lines
  • Determining the optimal solution to a system of linear inequalities
  • Solving systems of linear equations in physics and engineering

Tips and Tricks

Here are some tips and tricks to keep in mind when using the elimination method:

  • Make sure to multiply the equations by the necessary multiples to eliminate the variable.
  • Add the equations carefully to avoid making mistakes.
  • Check your work by plugging the solution back into the original equations.

Conclusion

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 one of the variables.

Q: When should I use the elimination method?

A: You should use the elimination method when the coefficients of the variables in the equations are relatively simple, and you want to eliminate one of the variables.

Q: How do I know which variable to eliminate?

A: You can eliminate either variable, but it's often easier to eliminate the variable with the larger coefficient.

Q: What if the coefficients of the variables are not the same?

A: If the coefficients of the variables are not the same, you can multiply the equations by the necessary multiples to make the coefficients the same.

Q: Can I use the elimination method with more than two equations?

A: Yes, you can use the elimination method with more than two equations, but it can become more complicated.

Q: What if I make a mistake while using the elimination method?

A: If you make a mistake while using the elimination method, you may end up with an incorrect solution. To avoid this, make sure to check your work by plugging the solution back into the original equations.

Q: Can I use the elimination method with systems of linear inequalities?

A: No, the elimination method is only used to solve systems of linear equations, not systems of linear inequalities.

Q: What are some common mistakes to avoid when using the elimination method?

A: Some common mistakes to avoid when using the elimination method include:

  • Not multiplying the equations by the necessary multiples
  • Not adding the equations carefully
  • Not checking the work by plugging the solution back into the original equations

Q: How do I know if the elimination method is the best method to use?

A: The elimination method is the best method to use when the coefficients of the variables are relatively simple, and you want to eliminate one of the variables. However, if the coefficients are complex, or if you want to find the solution graphically, you may want to use a different method.

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

A: Yes, you can use the elimination method with systems of linear equations with fractions, but you may need to multiply the equations by the necessary multiples to eliminate the fractions.

Q: What are some real-world applications of the elimination method?

A: Some real-world applications of the elimination method include:

  • Finding the intersection point of two lines
  • Determining the optimal solution to a system of linear inequalities
  • Solving systems of linear equations in physics and engineering

Conclusion

In conclusion, the elimination method is a powerful technique for solving systems of linear equations. By following the steps outlined in this article, and by avoiding common mistakes, you can successfully solve systems of linear equations and apply the method to real-world problems.