Eliminating The X Variable In Systems Of Equations A Step-by-Step Guide

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In the realm of algebra, solving systems of equations is a fundamental skill. Often, we encounter systems where one variable obstructs our path to a solution. This article delves into the art of eliminating the x variable from a system of equations, providing a step-by-step guide and practical examples to master this technique. Let's embark on a journey to unravel the intricacies of equation manipulation and variable elimination.

Understanding the Essence of Variable Elimination

Variable elimination is a powerful technique for solving systems of equations. The core idea revolves around manipulating the equations in such a way that one of the variables cancels out, leaving us with a single equation in one variable. This simplified equation can then be easily solved, and the value of the remaining variable can be substituted back into the original equations to find the value of the eliminated variable. This method is particularly useful when dealing with systems of linear equations, where the equations represent straight lines, and the solution corresponds to the point of intersection.

In the context of our problem, we have the following system of equations:

  1. 4x + 5y = 62
  2. 5x - 3y = 22

Our mission is to identify the constants by which each equation can be multiplied to eliminate the x variable. This involves careful consideration of the coefficients of x in both equations. By strategically multiplying each equation by a suitable constant, we can create coefficients of x that are additive inverses, meaning they have the same magnitude but opposite signs. When we add the modified equations together, the x terms will cancel out, leaving us with an equation solely in terms of y.

Key Concepts in Variable Elimination

Before we dive into the specifics of our problem, let's solidify our understanding of the key concepts involved in variable elimination:

  • Additive Inverses: Two numbers are additive inverses if their sum is zero. For instance, 5 and -5 are additive inverses. In the context of variable elimination, we aim to create coefficients of the variable we want to eliminate that are additive inverses.
  • Multiplication Property of Equality: This property states that if we multiply both sides of an equation by the same non-zero constant, the equation remains balanced. This is a crucial tool in variable elimination as it allows us to manipulate the coefficients of the variables without altering the solution of the system.
  • Combining Equations: Once we have manipulated the equations to create additive inverses for the coefficients of the variable we want to eliminate, we can add the equations together. This process effectively cancels out the variable, leaving us with a simpler equation.

The Art of Choosing the Right Constants

The crux of variable elimination lies in selecting the appropriate constants to multiply each equation by. The goal is to make the coefficients of the variable we want to eliminate additive inverses. In our case, we want to eliminate x, so we need to focus on the coefficients of x in the two equations: 4 and 5.

To make these coefficients additive inverses, we need to find a common multiple of 4 and 5. The least common multiple (LCM) of 4 and 5 is 20. This means we want to transform the coefficients of x into 20 and -20 (or vice versa). To achieve this, we can multiply the first equation by 5 and the second equation by -4.

Let's walk through the steps:

  1. Multiply the first equation by 5: 5 * (4x + 5y) = 5 * 62 20x + 25y = 310

  2. Multiply the second equation by -4: -4 * (5x - 3y) = -4 * 22 -20x + 12y = -88

Notice that the coefficients of x are now 20 and -20, which are additive inverses. This sets the stage for eliminating x when we combine the equations.

A Deeper Dive into Constant Selection

While the LCM method is a reliable approach, there are other ways to think about selecting the constants. We could also have multiplied the first equation by -5 and the second equation by 4. This would have resulted in coefficients of x being -20 and 20, which are also additive inverses. The key is to recognize that we need to find multiples of the coefficients that have opposite signs.

Furthermore, we could have chosen to eliminate y instead of x. In that case, we would have focused on the coefficients of y, which are 5 and -3. The LCM of 5 and 3 is 15. To make the coefficients of y additive inverses, we could multiply the first equation by 3 and the second equation by 5. This would result in coefficients of y being 15 and -15.

The flexibility in choosing which variable to eliminate and the constants to use highlights the versatility of variable elimination. It's a technique that can be adapted to suit different systems of equations.

The Elimination Process Unveiled

Now that we have strategically chosen our constants, let's proceed with the elimination process. We have transformed our original system of equations into:

  1. 20x + 25y = 310
  2. -20x + 12y = -88

To eliminate x, we simply add the two equations together. This is because the 20x and -20x terms will cancel each other out:

(20x + 25y) + (-20x + 12y) = 310 + (-88)

Combining like terms, we get:

37y = 222

Now we have a single equation in one variable, y. To solve for y, we divide both sides of the equation by 37:

y = 222 / 37 y = 6

We have successfully found the value of y! Now, we can substitute this value back into either of the original equations to solve for x. Let's use the first original equation:

4x + 5y = 62

Substitute y = 6:

4x + 5(6) = 62 4x + 30 = 62

Subtract 30 from both sides:

4x = 32

Divide both sides by 4:

x = 8

Therefore, the solution to the system of equations is x = 8 and y = 6.

Verification: A Crucial Step

To ensure the accuracy of our solution, it's always a good practice to verify it by substituting the values of x and y back into both original equations. Let's do that:

  1. First equation: 4x + 5y = 62 4(8) + 5(6) = 62 32 + 30 = 62 62 = 62 (True)

  2. Second equation: 5x - 3y = 22 5(8) - 3(6) = 22 40 - 18 = 22 22 = 22 (True)

Since our solution satisfies both original equations, we can confidently conclude that x = 8 and y = 6 is the correct solution.

Generalizing the Approach

The method we employed to eliminate x in this specific system of equations can be generalized to solve any system of linear equations with two variables. The general steps are:

  1. Identify the variable to eliminate: Choose the variable that appears easier to eliminate based on the coefficients in the equations.
  2. Find the LCM: Determine the least common multiple of the coefficients of the variable you want to eliminate.
  3. Multiply the equations: Multiply each equation by a constant such that the coefficients of the chosen variable become additive inverses (multiples of the LCM with opposite signs).
  4. Add the equations: Add the modified equations together to eliminate the chosen variable. This will result in a single equation in one variable.
  5. Solve for the remaining variable: Solve the equation obtained in step 4 to find the value of the remaining variable.
  6. Substitute back: Substitute the value obtained in step 5 back into either of the original equations to solve for the eliminated variable.
  7. Verify the solution: Substitute the values of both variables back into the original equations to ensure they are satisfied.

When Elimination Might Not Be the Best Choice

While variable elimination is a powerful technique, it's not always the most efficient method for solving systems of equations. In certain situations, other methods like substitution or graphing might be more suitable. For instance, if one of the equations is already solved for one variable in terms of the other, substitution might be a quicker approach. Similarly, if you need a visual representation of the solution, graphing the equations might be a better choice.

Practical Applications of Variable Elimination

Variable elimination is not just a theoretical concept confined to textbooks; it has numerous practical applications in various fields, including:

  • Engineering: Solving systems of equations is crucial in circuit analysis, structural analysis, and control systems design.
  • Economics: Economists use systems of equations to model supply and demand, market equilibrium, and macroeconomic relationships.
  • Computer Science: Systems of equations arise in areas like computer graphics, optimization algorithms, and cryptography.
  • Physics: Many physical phenomena, such as motion, forces, and energy, can be described using systems of equations.

By mastering variable elimination, you equip yourself with a valuable tool for tackling a wide range of problems in diverse fields.

Conclusion: The Power of Elimination

In this comprehensive guide, we have explored the art of eliminating the x variable from a system of equations. We have delved into the underlying principles, step-by-step procedures, and practical applications of this technique. By understanding the essence of variable elimination, mastering the art of choosing the right constants, and practicing the elimination process, you can confidently solve a wide range of systems of equations.

Remember, the key to success lies in understanding the concepts, practicing diligently, and verifying your solutions. With perseverance and a solid grasp of the principles, you can unlock the power of elimination and conquer the world of algebra!

Specifically, for the given system of equations:

  • 4x + 5y = 62
  • 5x - 3y = 22

We determined that multiplying the first equation by 5 and the second equation by -4 would eliminate the x variable. This is because it results in coefficients of x that are additive inverses (20 and -20). This understanding is not just about solving this particular problem; it's about developing a deeper understanding of the underlying principles that govern variable elimination.

By mastering these principles, you can approach any system of equations with confidence and the ability to choose the most efficient method for finding the solution. This skill is invaluable not only in mathematics but also in a wide range of fields where problem-solving and analytical thinking are essential. So, embrace the power of elimination and continue your journey to mathematical mastery!