Solving Systems Of Equations With Linear Combination The Elimination Method

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In the realm of mathematics, solving systems of equations is a fundamental skill with applications spanning various fields, from engineering and physics to economics and computer science. One of the most powerful techniques for tackling these systems is the linear combination method, also known as the elimination method. This method provides a systematic approach to eliminate one of the variables, simplifying the system and paving the way for a solution. This comprehensive guide delves into the intricacies of the linear combination method, illustrating its application with a detailed example and exploring its broader significance in mathematical problem-solving.

Understanding the Linear Combination Method

The linear combination method hinges on the principle of manipulating equations within a system to create opposite coefficients for one of the variables. By adding these modified equations, the targeted variable is eliminated, leaving a single equation with a single variable. This simplified equation can then be easily solved, and the solution can be back-substituted into one of the original equations to determine the value of the other variable. Let's break down the key steps involved in this process:

  1. Align the Equations: Begin by aligning the equations in the system, ensuring that like terms (terms with the same variable) are arranged in columns. This organization facilitates the subsequent steps.
  2. Multiply to Create Opposite Coefficients: Identify a variable to eliminate. Examine the coefficients of this variable in both equations. If the coefficients are not opposites, multiply one or both equations by constants to make them opposites. The goal is to have coefficients that, when added, will result in zero.
  3. Add the Equations: Once the coefficients of the chosen variable are opposites, add the equations together. This addition will eliminate the variable with opposite coefficients, resulting in a new equation with only one variable.
  4. Solve for the Remaining Variable: Solve the new equation for the remaining variable. This step typically involves basic algebraic manipulations, such as isolating the variable.
  5. Back-Substitute to Find the Other Variable: Substitute the value obtained in the previous step back into one of the original equations (or any equation in the system). Solve this equation for the other variable.
  6. Verify the Solution: To ensure accuracy, substitute both values obtained into the original equations. If both equations hold true, the solution is correct.

A Step-by-Step Example

Let's solidify our understanding with a concrete example. Consider the following system of equations:

x - y = -4
x + 5y = 59

Our objective is to find the values of x and y that satisfy both equations simultaneously. We can employ the linear combination method to achieve this.

  1. Align the Equations: The equations are already aligned, with x terms, y terms, and constants in their respective columns.

  2. Multiply to Create Opposite Coefficients: We can choose to eliminate either x or y. Let's eliminate y. Observe that the coefficients of y are -1 and 5. To make them opposites, we can multiply the first equation by 5:

    5(x - y) = 5(-4)  =>  5x - 5y = -20

    Now, our system looks like this:

    5x - 5y = -20
x + 5y = 59

  3. Add the Equations: Adding the two equations eliminates y:

    (5x - 5y) + (x + 5y) = -20 + 59
6x = 39

  4. Solve for the Remaining Variable: Solve for x:

    6x = 39  =>  x = 39 / 6  =>  x = 6.5

  5. Back-Substitute to Find the Other Variable: Substitute the value of x (6.5) into one of the original equations. Let's use the first equation:

    6.  5 - y = -4  =>  -y = -4 - 6.5  =>  -y = -10.5  =>  y = 10.5

  6. Verify the Solution: Substitute x = 6.5 and y = 10.5 into both original equations:

    Equation 1: 6.5 - 10.5 = -4  (True)
Equation 2: 6.5 + 5(10.5) = 59  =>  6.5 + 52.5 = 59  (True)

    Both equations hold true, so our solution is correct.

Therefore, the solution to the system of equations is x = 6.5 and y = 10.5.

Detailed breakdown of the provided system

Let's analyze the system of equations provided in the original prompt:

$egin{array}{ll} x-y=-4 & ightarrow 5(x-y)=5(-4) ightarrow 5x-5y=-20
x+5 y=59 & ightarrow x+5y=59

\end{array}$

Step 1: Multiply the first equation by 5

We start with the system:

  • x - y = -4
  • x + 5y = 59

To eliminate 'y', we can multiply the first equation by 5. This gives us:

  • 5x - 5y = -20
  • x + 5y = 59

Step 2: Add the equations

Now, add the modified first equation to the second equation:

(5x - 5y) + (x + 5y) = -20 + 59 This simplifies to:

  • 6x = 39

Step 3: Solve for x

Divide both sides by 6 to solve for x:

  • x = 39 / 6
  • x = 6.5

Step 4: Substitute x back into one of the original equations

Let's use the first original equation:

  • x - y = -4 Substitute x = 6.5:

    1. 5 - y = -4 Solve for y:

  • -y = -4 - 6.5

  • -y = -10.5

  • y = 10.5

So, the solution to the system is x = 6.5 and y = 10.5.

The Power and Versatility of Linear Combination

The linear combination method is a cornerstone of solving systems of linear equations due to its versatility and efficiency. It can be applied to systems with two, three, or even more variables, provided that the number of equations matches the number of variables. This method is particularly advantageous when dealing with systems where the coefficients of one variable are easily manipulated to become opposites.

Moreover, the linear combination method provides a clear and structured approach to problem-solving. Its step-by-step nature reduces the likelihood of errors and promotes a deeper understanding of the underlying mathematical principles. By mastering this technique, students and practitioners alike can confidently tackle a wide range of mathematical challenges.

Applications Beyond the Classroom

The significance of the linear combination method extends far beyond the classroom. It is a fundamental tool in various fields, including:

  • Engineering: Solving systems of equations is crucial in structural analysis, circuit design, and control systems.
  • Physics: Many physics problems, such as those involving forces, motion, and energy, can be modeled and solved using systems of equations.
  • Economics: Economic models often involve systems of equations that describe the relationships between different economic variables.
  • Computer Science: Systems of equations arise in areas such as linear programming, optimization, and computer graphics.

By understanding and applying the linear combination method, individuals can gain a valuable skill set that is highly sought after in numerous professions.

Conclusion

The linear combination method stands as a testament to the power of systematic problem-solving in mathematics. Its ability to eliminate variables and simplify complex systems makes it an indispensable tool for anyone working with equations. Whether you are a student grappling with algebraic concepts or a professional tackling real-world challenges, mastering the linear combination method will undoubtedly enhance your mathematical prowess and problem-solving capabilities. By following the steps outlined in this guide and practicing with diverse examples, you can confidently navigate the world of systems of equations and unlock their vast potential.