Solving Linear Equations: A Step-by-Step Guide

Let's dive into the world of algebra and tackle a common problem: solving linear equations. In this article, we'll break down the process step by step, making it easy to understand and apply. We'll use the example equation 3x + 2(4x - 6) = 8x + 1 to illustrate each stage. Whether you're a student brushing up on your skills or just curious about math, this guide is for you!

Understanding Linear Equations

Before we jump into solving, let's understand what a linear equation is. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are called "linear" because when graphed, they form a straight line. Solving a linear equation means finding the value of the variable (usually 'x') that makes the equation true. This involves isolating the variable on one side of the equation. The general form of a linear equation is ax + b = c, where a, b, and c are constants, and x is the variable. The key to solving these equations lies in performing the same operations on both sides to maintain balance. Whether it’s addition, subtraction, multiplication, or division, consistency is crucial. Remember, the goal is to get x by itself on one side, revealing its value and thus the solution to the equation. By mastering the techniques for solving linear equations, you build a foundational skill applicable to more complex algebraic problems and real-world mathematical applications. It's more than just manipulating numbers; it's about understanding the underlying structure of equations and how to logically isolate the unknown. So, let’s proceed with confidence and unlock the mystery behind linear equations, one step at a time.

Step 1: Distribute

The first step in solving our equation, 3x + 2(4x - 6) = 8x + 1, is to simplify it by distributing any terms. In this case, we need to distribute the '2' across the terms inside the parentheses (4x - 6). This means multiplying '2' by both '4x' and '-6'. So, 2 * 4x equals 8x, and 2 * -6 equals -12. After performing this distribution, our equation now looks like this: 3x + 8x - 12 = 8x + 1. Distributing terms is a crucial step because it removes parentheses, allowing us to combine like terms more easily. It's like unpacking a box to see what's inside before you start organizing things. Correctly applying the distributive property ensures that each term within the parentheses is properly accounted for, maintaining the equality of the equation. This step is not just about removing parentheses; it's about transforming the equation into a more manageable form where we can clearly see and combine similar terms. By mastering distribution, you’re setting yourself up for success in the subsequent steps of solving the equation. Remember, accuracy in this initial phase is vital to avoid errors that could propagate through the rest of the solution. So, double-check your multiplication and signs to ensure you’re on the right track.

Step 2: Combine Like Terms

After distributing, the next step is to combine like terms on each side of the equation. Looking at our equation, 3x + 8x - 12 = 8x + 1, we can see that there are two 'x' terms on the left side: '3x' and '8x'. These are like terms because they both contain the same variable, 'x', raised to the same power (which is 1 in this case). To combine them, we simply add their coefficients (the numbers in front of the 'x'). So, 3x + 8x equals 11x. Now, our equation looks like this: 11x - 12 = 8x + 1. Combining like terms simplifies the equation, making it easier to isolate the variable. It's like sorting your belongings into categories before organizing them – it makes the whole process more efficient. By reducing the number of terms, we make the equation less cluttered and more manageable. This step is essential for streamlining the solving process and reducing the chance of errors. Always double-check that you are only combining terms that have the exact same variable and exponent. Mixing up terms can lead to incorrect solutions. Mastering the art of combining like terms is a fundamental skill in algebra, enabling you to tackle more complex equations with confidence. So, take your time, be precise, and ensure that you’ve accurately combined all like terms on each side of the equation.

Step 3: Isolate the Variable Term

Now, we want to isolate the variable term, which means getting all the 'x' terms on one side of the equation and all the constant terms on the other side. Currently, our equation is 11x - 12 = 8x + 1. To get the 'x' terms on one side, we can subtract '8x' from both sides of the equation. This will eliminate the '8x' term on the right side and move it to the left side. So, 11x - 8x equals 3x. Subtracting '8x' from both sides, we get: 3x - 12 = 1. Isolating the variable term is a key step in solving for 'x' because it brings us closer to having 'x' by itself. It's like separating the ingredients you need for a recipe from the rest of your pantry. By strategically adding or subtracting terms from both sides, we maintain the balance of the equation while moving closer to our goal. Remember, whatever you do to one side of the equation, you must do to the other side to keep it equal. This principle is the foundation of solving equations. Double-check your arithmetic to avoid errors and ensure that you’re accurately moving the variable terms to one side. With each step, we’re refining the equation and bringing 'x' into sharper focus, setting the stage for the final solution.

Step 4: Isolate the Variable

Next, we need to completely isolate the variable 'x'. Our equation currently looks like this: 3x - 12 = 1. To isolate 'x', we must get rid of the '-12' on the left side. We can do this by adding '12' to both sides of the equation. Adding '12' to both sides gives us: 3x = 13. Now, 'x' is almost completely isolated. To finish the job, we need to get rid of the '3' that's multiplying 'x'. We can do this by dividing both sides of the equation by '3'. Dividing both sides by '3' gives us: x = 13/3. Isolating the variable is the final act in our equation-solving drama. It’s the moment where 'x' stands alone, revealing its true value. By carefully adding, subtracting, multiplying, or dividing, we strip away all the surrounding terms until 'x' is by itself. This step requires precision and attention to detail, ensuring that each operation is performed correctly on both sides of the equation. Remember, the goal is to undo any operations that are being applied to 'x' until it is completely isolated. This process is like peeling away layers to reveal the core – the value of 'x'. By mastering this final step, you gain the power to solve a wide range of linear equations and confidently uncover the hidden values of variables.

Step 5: Solution

Therefore, the solution to the equation 3x + 2(4x - 6) = 8x + 1 is x = 13/3. This means that if we substitute '13/3' for 'x' in the original equation, both sides of the equation will be equal. To check our answer, we can plug '13/3' back into the original equation: 3*(13/3) + 2*(4*(13/3) - 6) = 8*(13/3) + 1. Simplifying this expression, we get: 13 + 2*((52/3) - 6) = (104/3) + 1. Further simplification gives us: 13 + 2*(34/3) = (104/3) + 1. Which simplifies to: 13 + (68/3) = (104/3) + 1. Combining the terms, we get: (39/3) + (68/3) = (104/3) + (3/3). Finally, we have: (107/3) = (107/3), which confirms that our solution is correct. The satisfaction of finding the correct solution is one of the rewards of mastering algebra. It’s like completing a puzzle and seeing all the pieces fit perfectly together. By checking your answer, you ensure that your solution is accurate and that you have a solid understanding of the equation-solving process. This final step reinforces your confidence and demonstrates your ability to apply algebraic principles effectively. So, take the time to verify your solutions, and enjoy the satisfaction of knowing you’ve conquered the equation.

Conclusion

Solving linear equations is a fundamental skill in algebra. By following these steps – distributing, combining like terms, isolating the variable term, and isolating the variable – you can confidently solve a wide range of equations. Remember to always double-check your work to ensure accuracy. With practice, you'll become more proficient and comfortable with these techniques. Happy solving! For further learning about linear equations, visit Khan Academy's Linear Equations Section.