Eliminating Y In A System Of Equations A Step By Step Guide

by Scholario Team 60 views

In the realm of algebra, solving systems of equations is a fundamental skill. One common technique for solving these systems is elimination, where we manipulate the equations to eliminate one variable, making it easier to solve for the other. This article delves into the process of eliminating the variable y from a given system of two linear equations. We will explore the steps involved, the underlying mathematical principles, and provide a detailed explanation of why a specific operation is required to achieve this elimination.

Understanding the Equations

Before we dive into the elimination process, let's first examine the two equations we are working with:

-3x - 3y = 3 (E1)

-x - 2y = 1 (E2)

Our goal is to find a way to combine these two equations such that the y terms cancel each other out. This will leave us with a single equation in terms of x, which we can then easily solve.

The Elimination Strategy: Finding the Right Operation

To eliminate y, we need to make the coefficients of y in both equations equal in magnitude but opposite in sign. This way, when we add the equations together, the y terms will cancel out. Looking at our equations, the coefficients of y are -3 in E1 and -2 in E2. The least common multiple (LCM) of 3 and 2 is 6. Therefore, we want to manipulate the equations so that the coefficients of y become -6 and +6 (or vice versa).

To achieve this, we can multiply the first equation (E1) by 2 and the second equation (E2) by 3. This will give us:

2 * ( -3x - 3y ) = 2 * 3 => -6x - 6y = 6

3 * ( -x - 2y ) = 3 * 1 => -3x - 6y = 3

Now, we have two new equations:

-6x - 6y = 6

-3x - 6y = 3

Notice that the coefficients of y are both -6. To eliminate y, we need one of them to be +6. So, we can multiply the first equation (E1) by 2 and the second equation (E2) by 3. This results in the y coefficients being -6 in both modified equations. To effectively eliminate y, we require coefficients that are equal in magnitude but opposite in sign. Consequently, we should consider multiplying the first equation by 2 and the second equation by 3, which maintains the negative sign. Subsequently, we subtract the second modified equation from the first. This can be represented as 2E1 - 3E2. Let’s break down why this is the correct operation.

Step-by-Step Elimination: Why 2E1 - 3E2 Works

Let's perform the operation 2E1 - 3E2 step-by-step:

  1. Multiply E1 by 2:

    2 * (-3x - 3y) = 2 * 3 => -6x - 6y = 6

  2. Multiply E2 by 3:

    3 * (-x - 2y) = 3 * 1 => -3x - 6y = 3

  3. Subtract the modified E2 from the modified E1:

    (-6x - 6y) - (-3x - 6y) = 6 - 3

    -6x - 6y + 3x + 6y = 3

    -3x = 3

As you can see, the y terms have been successfully eliminated, leaving us with a simple equation in terms of x. Solving for x, we get x = -1.

In summary, the operation 2E1 - 3E2 is the correct choice because it results in the y terms having the same coefficient, allowing them to cancel out when the equations are subtracted.

Analyzing the Incorrect Options

To further solidify our understanding, let's examine why the other options are incorrect:

  • A. 3E1 + 2E2: This operation would result in the following:
    • 3E1: -9x - 9y = 9
    • 2E2: -2x - 4y = 2
    • Adding them: -11x - 13y = 11. The y terms do not cancel out.
  • B. 3E1 - 2E2: This operation would result in the following:
    • 3E1: -9x - 9y = 9
    • 2E2: -2x - 4y = 2
    • Subtracting them: -7x - 5y = 7. The y terms do not cancel out.
  • D. 2E1 + 3E2: This operation would result in the following:
    • 2E1: -6x - 6y = 6
    • 3E2: -3x - 6y = 3
    • Adding them: -9x - 12y = 9. The y terms do not cancel out.

As we can see, none of these operations lead to the elimination of y. They either result in different coefficients for y or coefficients with the same sign, preventing them from canceling out upon addition or subtraction.

Key Takeaways for Successful Elimination

Eliminating variables in systems of equations is a crucial skill in algebra. To successfully eliminate a variable, keep these key takeaways in mind:

  • Identify the Target Variable: Determine which variable you want to eliminate.
  • Adjust Coefficients: Multiply one or both equations by constants so that the coefficients of the target variable are equal in magnitude.
  • Opposite Signs are Key: Ensure that the coefficients of the target variable have opposite signs. If they don't, multiply one of the equations by -1.
  • Add or Subtract: If the coefficients have opposite signs, add the equations. If they have the same sign, subtract the equations.
  • Solve for the Remaining Variable: After elimination, you'll have an equation with only one variable. Solve for that variable.
  • Substitute Back: Substitute the value you found back into one of the original equations to solve for the other variable.

By following these steps and understanding the underlying principles, you can confidently tackle systems of equations and master the art of elimination.

Common Mistakes to Avoid

While the elimination method is powerful, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Forgetting to Distribute: When multiplying an equation by a constant, remember to distribute the constant to every term in the equation, including the constant term on the right-hand side.
  • Incorrectly Adding/Subtracting: Pay close attention to the signs when adding or subtracting equations. A simple sign error can throw off the entire solution.
  • Not Eliminating Completely: Ensure that the coefficients of the target variable are exactly equal in magnitude (but opposite in sign) before adding or subtracting. Otherwise, the variable won't be fully eliminated.
  • Substituting into the Wrong Equation: After solving for one variable, substitute its value back into one of the original equations (or a correctly modified version). Substituting into an equation that contains an error will lead to an incorrect solution.
  • Skipping Steps: It's tempting to try to do the elimination process in your head, but it's best to write out each step clearly, especially when dealing with more complex systems of equations. This will help you avoid careless errors.

By being aware of these common mistakes, you can increase your accuracy and confidence when using the elimination method.

Conclusion: Mastering Elimination for Algebraic Success

In conclusion, the operation 2E1 - 3E2 is the necessary step to eliminate y from the given system of equations. This operation ensures that the coefficients of y become equal in magnitude and opposite in sign, allowing for their cancellation when the equations are combined. Mastering the elimination method is a crucial step in developing strong algebraic skills. By understanding the principles behind it and practicing regularly, you can confidently solve a wide range of systems of equations. Remember to pay close attention to the signs, distribute carefully, and double-check your work to avoid common mistakes. With dedication and practice, you'll become proficient in using elimination and other algebraic techniques to solve mathematical problems effectively.

This comprehensive guide has provided a detailed explanation of the elimination process, including the reasoning behind the correct operation, analysis of incorrect options, and key takeaways for success. By applying these principles, you can confidently tackle similar problems and enhance your algebraic skills.

By understanding the nuances of variable elimination and practicing consistently, you can build a solid foundation in algebra and excel in your mathematical endeavors. Remember that mathematics is a journey of learning and discovery, and each step you take brings you closer to mastery. Keep exploring, keep questioning, and keep practicing, and you'll unlock the power of algebra to solve complex problems and make meaningful connections in the world around you.

Practice Problems

To further reinforce your understanding of variable elimination, try solving these practice problems:

  1. Solve the system of equations:

    2x + y = 7

    x - y = 2

  2. Solve the system of equations:

    3x - 2y = 5

    x + y = 10

  3. Solve the system of equations:

    4x + 3y = 11

    2x - y = 1

By working through these problems, you'll gain valuable experience in applying the elimination method and build your problem-solving skills. Remember to break down each problem into steps, carefully perform the necessary operations, and double-check your work to ensure accuracy.

Further Exploration

If you're interested in learning more about systems of equations and variable elimination, there are many resources available online and in textbooks. You can explore different methods for solving systems of equations, such as substitution and graphing, and delve into more advanced topics, such as systems of linear inequalities and nonlinear systems of equations.

By continuously expanding your knowledge and skills, you'll become a more confident and capable mathematician. Remember that learning is a lifelong journey, and there's always something new to discover in the fascinating world of mathematics.