What Is The First Step To Solve The System Of Equations By Elimination?$\[ 6x + 3y = -12 \\]$\[ 6x + Y = 4 \\]A. Add The Two Equations To Eliminate The \[$ X \$\] Terms.B. Add The Two Equations To Eliminate The \[$ Y

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Introduction

Solving a system of equations is a fundamental concept in mathematics, and one of the most effective methods to solve it is by elimination. The elimination method involves adding or subtracting equations to eliminate one of the variables, making it easier to solve for the other variable. In this article, we will discuss the first step to solve the system of equations by elimination.

Understanding the System of Equations

A system of equations is a set of two or more equations that contain the same variables. In this case, we have two equations:

6x+3y=−12{ 6x + 3y = -12 }

6x+y=4{ 6x + y = 4 }

Our goal is to solve for the values of x and y that satisfy both equations.

The Elimination Method

The elimination method involves adding or subtracting equations to eliminate one of the variables. To eliminate a variable, we need to have the same coefficient for that variable in both equations. In this case, we can see that both equations have a coefficient of 6 for the variable x.

Step 1: Multiply the Equations by Necessary Multiples

To eliminate the x term, we need to have the same coefficient for x in both equations. We can do this by multiplying the second equation by 3, so that the coefficient of x in both equations is 18.

6x+3y=−12{ 6x + 3y = -12 }

18x+3y=12{ 18x + 3y = 12 }

Now, we can see that the coefficient of x in both equations is 18.

Step 2: Add the Two Equations to Eliminate the x Term

Now that we have the same coefficient for x in both equations, we can add the two equations to eliminate the x term.

(6x+3y)+(18x+3y)=(−12)+(12){ (6x + 3y) + (18x + 3y) = (-12) + (12) }

24x+6y=0{ 24x + 6y = 0 }

By adding the two equations, we have eliminated the x term.

Conclusion

In conclusion, the first step to solve the system of equations by elimination is to multiply the equations by necessary multiples to have the same coefficient for the variable to be eliminated. In this case, we multiplied the second equation by 3 to have the same coefficient for x in both equations. Then, we added the two equations to eliminate the x term. This is the first step in solving the system of equations by elimination.

Why is the Elimination Method Important?

The elimination method is an important concept in mathematics because it allows us to solve systems of equations in a straightforward and efficient manner. By eliminating one of the variables, we can make it easier to solve for the other variable. This method is particularly useful when dealing with systems of linear equations.

Real-World Applications of the Elimination Method

The elimination method has many real-world applications, including:

  • Physics and Engineering: The elimination method is used to solve systems of equations that describe the motion of objects in physics and engineering.
  • Economics: The elimination method is used to solve systems of equations that describe the behavior of economic systems.
  • Computer Science: The elimination method is used to solve systems of equations that describe the behavior of computer systems.

Common Mistakes to Avoid

When using the elimination method, there are several common mistakes to avoid:

  • Not having the same coefficient for the variable to be eliminated: Make sure to multiply the equations by necessary multiples to have the same coefficient for the variable to be eliminated.
  • Not adding the equations correctly: Make sure to add the equations correctly to eliminate the variable.
  • Not checking the solution: Make sure to check the solution to ensure that it satisfies both equations.

Conclusion

Q: What is the elimination method?

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

Q: Why is the elimination method important?

A: The elimination method is important because it allows us to solve systems of equations in a straightforward and efficient manner. By eliminating one of the variables, we can make it easier to solve for the other variable.

Q: What are the steps involved in the elimination method?

A: The steps involved in the elimination method are:

  1. Multiply the equations by necessary multiples to have the same coefficient for the variable to be eliminated.
  2. Add the equations to eliminate the variable.
  3. Solve for the remaining variable.

Q: How do I know which variable to eliminate?

A: To determine which variable to eliminate, look for the variable that has the same coefficient in both equations. This variable can be eliminated by adding or subtracting the equations.

Q: What if the coefficients of the variable to be eliminated are not the same?

A: If the coefficients of the variable to be eliminated are not the same, multiply the equations by necessary multiples to have the same coefficient for the variable to be eliminated.

Q: Can I use the elimination method to solve systems of equations with more than two variables?

A: Yes, the elimination method can be used to solve systems of equations with more than two variables. However, it may be more complicated and require more steps.

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 having the same coefficient for the variable to be eliminated.
  • Not adding the equations correctly.
  • Not checking the solution.

Q: How do I check the solution to ensure it satisfies both equations?

A: To check the solution, substitute the values of the variables into both equations and verify that they are true.

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

A: Yes, the elimination method can be used to solve systems of equations with fractions or decimals. However, it may be more complicated and require more steps.

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

A: The elimination method has many real-world applications, including:

  • Physics and engineering: The elimination method is used to solve systems of equations that describe the motion of objects.
  • Economics: The elimination method is used to solve systems of equations that describe the behavior of economic systems.
  • Computer science: The elimination method is used to solve systems of equations that describe the behavior of computer systems.

Conclusion

In conclusion, the elimination method is a powerful tool for solving systems of equations. By following the steps outlined in this article and avoiding common mistakes, you can use the elimination method to solve systems of equations in a straightforward and efficient manner. Remember to multiply the equations by necessary multiples to have the same coefficient for the variable to be eliminated, and then add the equations to eliminate the variable. With practice and patience, you can become proficient in using the elimination method to solve systems of equations.