Solving Linear Equations With Elimination Method Step-by-Step Guide

Hey guys, struggling with systems of linear equations? Don't worry, we've all been there! If you're facing a problem like this: 2x + 6y = 10 and 4x - 3y = -10, and your teacher wants you to solve it using the elimination method, you've come to the right place. This method is super useful for getting rid of one variable at a time, making the whole process way easier. Let's break it down step by step, so you can ace that homework or test!

What is the Elimination Method?

The elimination method is a technique used to solve systems of linear equations by eliminating one of the variables. This is achieved by manipulating the equations so that the coefficients of one variable are opposites (e.g., 2 and -2). When you add the equations together, that variable cancels out, leaving you with a single equation in one variable. This resulting equation can then be easily solved, and the value of the solved variable can be substituted back into either of the original equations to find the value of the other variable. Think of it as a strategic way to simplify complex equations, making them much easier to handle. The beauty of the elimination method lies in its directness and efficiency, especially when dealing with equations where coefficients are easily manipulated to create opposites. It's a core technique in algebra, and mastering it will definitely boost your problem-solving skills.

Before we dive into the problem at hand, let's really understand why the elimination method is so powerful. In essence, it allows us to strategically simplify complex systems of equations. Imagine trying to solve for two unknowns, x and y, simultaneously. It can feel like a puzzle with too many pieces. The elimination method gives us a way to reduce that puzzle, getting rid of one piece at a time. We manipulate the equations, usually by multiplying one or both by a constant, so that either the x or y terms have the same coefficient but opposite signs. This is the crucial setup. When we then add the equations together, these terms cancel each other out—they're eliminated! We're left with a single equation with just one unknown. Boom! We can solve for that variable. Once we've found the value of one variable, the rest is relatively straightforward. We substitute that value back into any of the original equations (or even the modified ones) and solve for the other variable. This process, step-by-step, transforms a seemingly daunting problem into a series of manageable steps. The elimination method isn't just a trick; it's a powerful tool rooted in fundamental algebraic principles. It relies on the idea that if you perform the same operation on both sides of an equation, you maintain the equality. By strategically using multiplication and addition, we can systematically isolate variables and find solutions. So, next time you face a system of equations, remember the elegance and efficiency of elimination. It's your secret weapon for conquering algebraic challenges.

Solving 2x + 6y = 10 and 4x - 3y = -10

Okay, let's tackle the problem: 2x + 6y = 10 and 4x - 3y = -10. Our goal is to find the values of x and y that satisfy both equations. Here’s how we can do it using elimination:

Step 1: Choose a Variable to Eliminate

Look at the coefficients of x and y in both equations. Notice that the y coefficients (6 and -3) are easier to manipulate to become opposites. We can multiply the second equation by 2 to make the y coefficient -6, which is the opposite of 6 in the first equation. Choosing which variable to eliminate first is a strategic decision that can significantly impact the ease of solving the system. Generally, it’s wise to look for coefficients that are multiples of each other or that have opposite signs, as these will require fewer manipulations to set up for elimination. In this case, focusing on the y-variable is a smart move because the coefficients 6 and -3 are conveniently related. Multiplying the second equation by 2 is a straightforward step that will create the necessary opposite coefficients. But why is this strategic choice so important? Well, consider the alternative: eliminating x first. We’d need to multiply the first equation by -2, a single, relatively simple operation. However, focusing on y sets us up for an even smoother process. By doubling the second equation, we create the -6y term, perfectly poised to cancel out the +6y term in the first equation. This minimizes the chances of making arithmetic errors and streamlines the solution process. The ability to spot these opportunities for efficient elimination is a key skill in algebra. It transforms what might seem like a complex problem into a series of clear, manageable steps. So, when you're faced with a system of equations, take a moment to survey the coefficients, look for those easy relationships, and choose your elimination target wisely. Your future self will thank you!

Step 2: Multiply the Equations

Multiply the second equation (4x - 3y = -10) by 2. This gives us a new equation: 8x - 6y = -20. We keep the first equation as it is: 2x + 6y = 10. Multiplying an equation by a constant is a fundamental technique in the elimination method, and it's crucial to understand why it works and how it sets us up for success. The core idea is that we're not changing the solution of the equation; we're simply rewriting it in a way that makes it easier to work with. When we multiply every term in the equation (both sides!) by the same constant, we maintain the equality. Think of it like scaling a recipe: if you double all the ingredients, you still get the same dish, just a larger portion. In the context of solving systems of equations, this scaling allows us to strategically manipulate the coefficients of the variables. We choose a constant that will make the coefficients of one variable opposites in the two equations. This is the magic that allows for elimination. In our case, multiplying the second equation by 2 turns the -3y term into -6y, which is the exact opposite of the +6y term in the first equation. Now, when we add the equations together, these y terms will cancel each other out, leaving us with an equation in just one variable. This is the power of strategic multiplication. It's not just about changing numbers; it's about reshaping the equations to reveal hidden simplifications. So, remember, when you multiply an equation in the elimination method, you're not just doing arithmetic; you're making a tactical move towards a solution.

Step 3: Add the Equations

Now, add the two equations together:

  2x + 6y = 10
+ 8x - 6y = -20
----------------
 10x + 0y = -10

This simplifies to 10x = -10. Adding equations together is the heart of the elimination method, the step where all the strategic manipulation we've done pays off. It's not just about adding numbers; it's about combining entire equations in a way that eliminates one of the variables. Think of it like adding two puzzle pieces together: when they fit perfectly, certain parts disappear, and a clearer picture emerges. In our case, we've carefully arranged the equations so that the y-terms have opposite coefficients. When we add the equations, these terms cancel each other out, leaving us with an equation that involves only x. This is the breakthrough moment. We've reduced a system of two equations with two unknowns to a single equation with one unknown, which we can easily solve. But why does this work? It's based on a fundamental property of equality: if we add equal quantities to both sides of an equation, the equality remains true. Since the left side of one equation equals the right side, and vice versa for the other equation, we can add the left sides together and the right sides together without changing the overall balance. The key is to add the equations in a way that eliminates a variable. This requires careful preparation, which is why we focus on multiplying equations in the earlier steps. We're setting up the puzzle pieces to fit together perfectly. So, when you add equations in the elimination method, remember that you're not just doing arithmetic; you're executing the core strategic move that transforms the entire system into a solvable form. It's the moment when all the preparation leads to a clear path forward.

Step 4: Solve for x

Divide both sides of 10x = -10 by 10 to get x = -1. Solving for a single variable after elimination is often the most straightforward part of the process, but it's also a crucial step that solidifies our progress and brings us closer to the complete solution. Once we've successfully eliminated one variable by adding the equations together, we're left with a simple equation involving only the other variable. This equation is usually in the form of 'ax = b' or 'ay = b', where 'a' and 'b' are constants. To isolate the variable, we perform the inverse operation of whatever is being done to it. In most cases, this involves division. We divide both sides of the equation by the coefficient of the variable we're solving for. The fundamental principle behind this is maintaining balance: whatever we do to one side of the equation, we must do to the other side to keep the equality intact. This ensures that the solution we find is valid. In our example, after adding the equations, we arrived at 10x = -10. To isolate x, we divide both sides by 10, resulting in x = -1. This seemingly simple step is a major milestone. We've determined the value of one of the unknowns, which is a significant piece of the puzzle. Now that we know x, we can move on to the next phase: substituting this value back into one of the original equations to find y. Solving for x (or y) after elimination is not just a mechanical process; it's a moment of clarity. It's the point where the strategic maneuvers we've made start to yield concrete results. It's a testament to the power of algebraic manipulation and a key step in our journey to solving the system of equations.

Step 5: Substitute x into One of the Original Equations

Plug x = -1 into the first equation: 2(-1) + 6y = 10. This simplifies to -2 + 6y = 10. Substituting the value of one variable back into one of the original equations is a pivotal step in solving systems of equations, and it's where we leverage the information we've gained to unlock the final piece of the puzzle. Once we've solved for one variable (let's say x) using elimination, we know its numerical value. But we still need to find the value of the other variable (y). This is where substitution comes in. We choose one of the original equations (it usually doesn't matter which one, but sometimes one might look simpler than the other) and replace the x-variable with the value we just found. This transforms the equation into one that contains only the y-variable. The beauty of this step is that it converts a system of two equations into a single equation with a single unknown. This equation can then be solved using standard algebraic techniques. In essence, substitution allows us to leverage the knowledge we've gained to progressively simplify the problem. We're not starting from scratch; we're building upon our previous work. By substituting the value of x, we're essentially reducing the problem's complexity, making it more manageable. The choice of which original equation to use for substitution is often a matter of convenience. Sometimes, one equation might have smaller coefficients or fewer terms, making the arithmetic a bit easier. But the core principle remains the same: we're using the known value of one variable to find the unknown value of the other. So, when you substitute, remember that you're not just plugging in a number; you're strategically using information to simplify the problem and move closer to the complete solution.

Step 6: Solve for y

Add 2 to both sides: 6y = 12. Then, divide both sides by 6: y = 2. Solving for the remaining variable after substitution is the final algebraic step in finding the solution to a system of equations, and it's where we bring all our efforts to a satisfying conclusion. After substituting the value of one variable (like x) into an original equation, we're left with a single equation containing only the other variable (y). This equation is our key to unlocking the final answer. The process of solving for y is similar to solving for x in the earlier steps: we use inverse operations to isolate the y-variable on one side of the equation. This might involve adding or subtracting constants from both sides, multiplying or dividing by coefficients, or even dealing with more complex expressions. The guiding principle is always to maintain balance: whatever operation we perform on one side of the equation, we must also perform on the other side to preserve the equality. This ensures that the value we find for y is a valid solution. The specific steps involved in solving for y will depend on the equation we obtained after substitution. However, the underlying strategy remains the same: we want to isolate y by undoing any operations that are being performed on it. Once we've successfully isolated y, we'll have its numerical value, which, combined with the value we found for x, gives us the complete solution to the system of equations. Solving for y is not just a mechanical process; it's the culmination of all our strategic maneuvers. It's the moment where the puzzle pieces finally fall into place, and we see the full picture. It's a testament to the power of algebraic thinking and a rewarding end to our problem-solving journey.

Step 7: Write the Solution

The solution is x = -1 and y = 2, or as an ordered pair, (-1, 2). Writing the solution to a system of equations in a clear and organized way is just as important as the algebraic steps we take to find it, because it ensures that our answer is easily understood and can be readily applied in other contexts. Once we've solved for both variables (x and y), we have the numerical values that satisfy both equations in the system. But simply stating these values isn't always the clearest way to present the solution. The most common and widely accepted way to express the solution is as an ordered pair, written in the form (x, y). This notation emphasizes that the solution is a pair of values that work together, with x always listed first and y second. The parentheses and comma are essential parts of this notation, as they clearly indicate that we're dealing with an ordered pair rather than two separate numbers. The ordered pair representation is particularly useful when we're dealing with systems of equations that represent lines on a graph. The solution (x, y) corresponds to the point where the two lines intersect. This visual interpretation makes the solution more intuitive and helps to connect algebra with geometry. In addition to the ordered pair notation, it's sometimes helpful to explicitly state the values of x and y separately, especially when explaining the solution process. This can make it easier for someone to follow our reasoning and see how we arrived at the answer. So, when you write the solution to a system of equations, remember that clarity and organization are key. Using the ordered pair notation and clearly stating the values of x and y will ensure that your answer is communicated effectively.

Checking Your Answer

It's always a good idea to check your answer. Plug x = -1 and y = 2 into both original equations:

• 2(-1) + 6(2) = -2 + 12 = 10 (Correct!)
• 4(-1) - 3(2) = -4 - 6 = -10 (Correct!)

Since the solution satisfies both equations, we know we've done it right!

Verifying your solution is crucial! It's like the final brushstroke on a masterpiece, ensuring that your hard work has truly paid off and that your answer is accurate. Plugging the values you've found back into the original equations serves as a robust check against potential errors you might have made during the algebraic manipulations. Think of it as a double-check, a way to confirm that the numbers you've calculated actually work within the context of the problem. The process is straightforward: you take the values you've found for x and y and substitute them into each of the original equations. Then, you simplify the expressions on both sides of the equation to see if they are equal. If they are, it's a strong indication that your solution is correct. If they aren't, it means there's a mistake somewhere in your calculations, and you need to go back and review your steps. Checking your answer is not just about getting the right answer; it's about developing good problem-solving habits. It fosters a sense of responsibility for your work and encourages you to be meticulous in your approach. It also helps you build confidence in your ability to solve problems accurately. Moreover, verification can be a valuable learning experience. If you find that your solution doesn't check, the process of tracing back your steps to identify the error can deepen your understanding of the concepts and techniques involved. So, always make checking your solution a standard part of your problem-solving routine. It's a small investment of time that can yield significant dividends in terms of accuracy, confidence, and learning.

Wrapping Up

The elimination method might seem tricky at first, but with practice, it becomes a powerful tool for solving systems of equations. Remember to choose the easiest variable to eliminate, multiply equations strategically, and always check your answer. You got this! This method is not just about finding answers; it's about developing a systematic approach to problem-solving that you can apply in many areas of mathematics and beyond. The ability to break down complex problems into smaller, more manageable steps, to identify patterns and relationships, and to think strategically about how to manipulate equations – these are skills that will serve you well in any field you pursue. The elimination method is a microcosm of the problem-solving process itself. It teaches you the importance of planning, of being organized, and of paying attention to detail. It shows you that persistence and patience are key to overcoming challenges. And it demonstrates the power of verification, of taking the time to check your work and ensure that your solution is correct. So, as you continue to practice and refine your skills, remember that you're not just learning a mathematical technique; you're developing a valuable mindset. You're becoming a more confident, capable problem-solver, ready to tackle whatever challenges come your way. The elimination method is a stepping stone on your journey to mathematical mastery, and the skills you acquire along the way will empower you to excel in all your future endeavors.