Solving For X: A Step-by-Step Guide

Are you ready to solve for x? Don't worry, it's easier than it looks! We're going to break down the equation $3x + 1 = -2x + 36$ step by step so you can understand how to find the value of x. This guide will help you grasp the fundamentals of algebraic equations and build your confidence in solving them. Whether you're a student struggling with homework or just brushing up on your math skills, this is the perfect place to start. We will go through each operation so you can understand the basics of the equation and it's logic. So grab your pen and paper, and let's get started. We will also include some helpful tips and tricks along the way to make your learning experience smooth and effective. By the end of this guide, you will be able to solve basic algebraic equations with ease. Let's make math fun and interesting. In the world of mathematics, understanding how to isolate a variable is a fundamental skill. And this skill is essential for solving a wide variety of problems, from simple equations to complex scientific formulas. The process typically involves a series of algebraic manipulations aimed at getting the target variable by itself on one side of the equation. This may involve adding, subtracting, multiplying, or dividing terms on both sides of the equation. The key principle is to maintain the balance of the equation by performing the same operations on both sides. This ensures that the equation remains valid throughout the solution process. One of the most important aspects is the careful use of inverse operations to undo operations that are applied to the variable. For example, to undo addition, you would use subtraction, and to undo multiplication, you would use division. Mastery of these techniques enables you to find the value of an unknown variable effectively. Furthermore, solving for x involves applying these techniques systematically, step by step, until you successfully isolate x. Let's get started!

Step 1: Combining Like Terms

The first step to solving for x is to combine like terms. This means getting all the x terms on one side of the equation and the constant numbers on the other side. This is like sorting your socks and shoes. We want all the x socks together and the number shoes together. In the equation $3x + 1 = -2x + 36$, we have an x term on both sides. To move the $-2x$ to the left side, we need to do the opposite operation: adding $2x$ to both sides. Remember, whatever we do to one side of the equation, we must do to the other side to keep it balanced. When we add $2x$ to both sides, the equation becomes $3x + 2x + 1 = -2x + 2x + 36$. Simplifying this gives us $5x + 1 = 36$. Next, we need to move the constant term, $+1$, to the right side of the equation. To do this, we subtract $1$ from both sides: $5x + 1 - 1 = 36 - 1$. This simplifies to $5x = 35$. Now, we have successfully combined our x terms and our constant terms. This step is about organizing our equation so that we can easily find the value of x. Combining like terms is a foundational skill in algebra. It helps simplify complex equations, making them easier to solve. The underlying principle is to gather similar terms on the same side of the equation to isolate the variable we're trying to find. This process involves recognizing and combining terms that have the same variable or are constant numbers. The key to mastering this step lies in understanding the concept of inverse operations. For example, if you have $+ 2x$ on one side of the equation, you can eliminate it by subtracting $2x$ from both sides. This is because adding and subtracting are inverse operations, meaning they undo each other. The ultimate goal here is to manipulate the equation until only the variable remains on one side, and the constant terms are on the other side. Through practice and careful attention to detail, you will quickly become proficient at combining like terms, setting the stage for more advanced algebraic manipulations. Remember, each step is like building a puzzle, and combining like terms is one of the essential pieces. The balance of the equation must be maintained at all times.

Step 2: Isolating the Variable

Now that we have $5x = 35$, the next step is to isolate the variable x. Right now, x is being multiplied by 5. To isolate x, we need to do the inverse operation of multiplication, which is division. We will divide both sides of the equation by 5. This gives us $\frac{5x}{5} = \frac{35}{5}$. On the left side, the 5s cancel out, leaving us with just x. On the right side, $35$ divided by $5$ is $7$. So, we have $x = 7$. Congratulations, we have solved for x! In simpler terms, to isolate the variable, you want to get x all by itself. Think of it like a treasure hunt; you want to find the hidden value of x. The equation is like a set of clues, and each operation you perform is a step closer to the treasure. This step involves getting rid of any coefficients or constants that are attached to x. Remember, whatever operation you do on one side of the equation, you must do it on the other side to maintain balance. This ensures that the equation remains valid. By diligently performing these inverse operations, you can gradually peel away all the extra terms and reveal the hidden value of x. Isolating the variable is a crucial part of algebra because it allows us to find unknown values and solve real-world problems. Whether you're balancing a checkbook, calculating the cost of groceries, or analyzing scientific data, your ability to isolate variables will serve you well. Remember, patience and practice are key to mastering this skill. Keep working through examples, and soon you'll be solving equations with confidence! Always check your work by substituting the value of x back into the original equation to ensure it's correct.

Step 3: Checking Your Answer

Always check your answer! This is a crucial step to ensure that your solution is correct. Substitute the value of x back into the original equation: $3x + 1 = -2x + 36$. We found that $x = 7$. So, substitute 7 for x in the original equation: $3(7) + 1 = -2(7) + 36$. Simplify: $21 + 1 = -14 + 36$. Further simplify: $22 = 22$. Since both sides of the equation are equal, our answer is correct! Always do this check to make sure that the solution you found is valid. Checking your answer is like the final test in a recipe; it confirms that you've followed all the steps correctly. This step is about verifying that your solution actually works in the original equation. It's a simple process, but it can save you from making careless mistakes. By substituting your value for x back into the original equation, you can easily determine whether your solution is accurate. If the equation holds true, then your answer is correct. If the equation does not balance, then you'll know that you need to go back and review your steps to find where you went wrong. Checking your answer helps to build your confidence in your math skills, knowing that you can solve the equations with accuracy. This practice also reinforces your understanding of the equation and the problem-solving process. Ultimately, taking the time to check your answer is an essential habit for any aspiring mathematician. It will help you avoid errors, improve your problem-solving skills, and feel more confident in your ability to solve equations correctly. It also gives you practice to work with the equations and helps memorize the operations and formula.

Conclusion: Mastering the Equation

We have successfully solved for x in the equation $3x + 1 = -2x + 36$. The steps were:

1. Combine Like Terms: Add $2x$ to both sides to get $5x + 1 = 36$, then subtract $1$ from both sides to get $5x = 35$.
2. Isolate the Variable: Divide both sides by $5$ to find $x = 7$.
3. Check Your Answer: Substitute $x = 7$ back into the original equation to verify the solution. Always remember to maintain the balance of the equation by performing the same operations on both sides. Practice these steps with different equations, and soon you'll be solving equations with confidence! Keep practicing, and you'll become a pro in no time! Solving for x is a foundational skill in algebra, providing the ability to solve a wide variety of problems. Mastering the process of isolating a variable not only enhances your mathematical proficiency but also cultivates problem-solving skills essential in various aspects of life. In solving for x, remember to stay organized, maintaining balance, using the right operations, and checking your answers. These are the cornerstones of successful problem-solving. Each equation you solve is a step towards building confidence in your mathematical capabilities. Embrace the challenge, and never hesitate to seek help when needed. The ability to solve for x is a valuable asset, opening doors to advanced mathematical concepts and real-world applications. Practice is key; the more you practice, the more comfortable and proficient you'll become. So, keep working through examples and applying what you've learned. With each equation you solve, you'll feel a sense of accomplishment and a growing understanding of mathematical principles. Celebrate your successes, learn from your mistakes, and most importantly, keep enjoying the journey of learning. It's important to remember that algebra builds on itself. The concepts that you learn now will be the building blocks for more advanced topics in the future. So, by mastering this basic equation, you're setting yourself up for success in more complex math problems.

For more information on the topic, you can visit Khan Academy at https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:solve-equations-inequalities This website provides comprehensive resources and interactive exercises to enhance your understanding of algebra and other mathematical concepts.