Solving 16x - 4 + 5x = -67 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of algebraic problem-solving. This article delves into the process of solving a specific linear equation, providing a comprehensive, step-by-step guide suitable for students and anyone looking to refresh their algebra skills. Our focus will be on the equation 16x - 4 + 5x = -67, a typical example that combines variable terms, constants, and the core principles of equation manipulation. By the end of this guide, you'll not only understand how to solve this particular equation but also grasp the broader techniques applicable to a wide range of linear equations. We'll break down each step, explaining the logic and mathematical rules involved, ensuring a clear and thorough understanding of the solution process. So, let's embark on this mathematical journey and unravel the mystery of finding the value of 'x'.

Understanding the Basics of Linear Equations

Before we tackle the equation 16x - 4 + 5x = -67, let's solidify our understanding of what a linear equation is. A linear equation is, at its core, a mathematical statement asserting the equality between two expressions. These expressions are composed of variables (usually denoted by letters like 'x', 'y', or 'z'), constants (numbers), and mathematical operations such as addition, subtraction, multiplication, and division. The term "linear" indicates that the highest power of the variable in the equation is one. This means we won't encounter terms like x², x³, or any other exponents greater than one. Understanding this fundamental characteristic is crucial because it dictates the methods we can use to solve the equation.

The goal in solving a linear equation is to isolate the variable on one side of the equation. This means we want to manipulate the equation until we have 'x' (or whatever variable we're solving for) alone on one side, and a numerical value on the other side. This numerical value then represents the solution to the equation – the value of 'x' that makes the equation true. The beauty of linear equations lies in their straightforward nature; they can be solved using a consistent set of algebraic principles. These principles involve performing the same operations on both sides of the equation to maintain balance, a concept we'll explore in detail as we solve our example equation. By mastering these basics, you lay a solid foundation for tackling more complex algebraic problems in the future.

Step 1: Combine Like Terms

The first step in solving the equation 16x - 4 + 5x = -67 is to simplify each side by combining like terms. Like terms are those that contain the same variable raised to the same power. In our equation, we have two terms containing 'x': 16x and 5x. These are like terms and can be combined. We also have a constant term, -4, which cannot be combined with the 'x' terms at this stage. To combine 16x and 5x, we simply add their coefficients (the numbers in front of the 'x'): 16 + 5 = 21. This gives us 21x.

Now, our equation looks like this: 21x - 4 = -67. Notice how combining like terms has made the equation simpler and more manageable. This is a crucial step in solving any equation, as it reduces the complexity and makes subsequent steps easier to perform. By combining like terms, we're essentially tidying up the equation, grouping similar elements together. This not only makes the equation visually clearer but also brings us closer to isolating the variable 'x'. Remember, the principle of combining like terms applies to any linear equation, regardless of its complexity. It's a foundational technique that will serve you well in your algebraic endeavors. The ability to quickly and accurately combine like terms is a key skill in simplifying equations and paving the way for a smooth solution process.

Step 2: Isolate the Variable Term

After combining like terms, our equation stands as 21x - 4 = -67. The next step in our quest to solve for 'x' is to isolate the variable term. The variable term, in this case, is 21x. To isolate it, we need to eliminate any constants that are on the same side of the equation. Here, we have the constant -4, which is being subtracted from 21x. To eliminate it, we perform the inverse operation: addition. We add 4 to both sides of the equation. This is a crucial step, as it maintains the balance of the equation. What we do to one side, we must do to the other.

Adding 4 to both sides, we get: 21x - 4 + 4 = -67 + 4. On the left side, -4 and +4 cancel each other out, leaving us with just 21x. On the right side, -67 + 4 equals -63. So, our equation now looks like this: 21x = -63. We've successfully isolated the variable term on the left side. This step highlights the fundamental principle of maintaining balance in an equation. By adding the same value to both sides, we ensure that the equality remains true. Isolating the variable term is a critical milestone in the solving process, as it brings us one step closer to determining the value of 'x'. This technique of using inverse operations is a cornerstone of algebra and will be used repeatedly in solving various types of equations.

Step 3: Solve for x

Having isolated the variable term, our equation now reads 21x = -63. The final step in solving for 'x' is to isolate 'x' completely. Currently, 'x' is being multiplied by 21. To undo this multiplication and isolate 'x', we perform the inverse operation: division. We divide both sides of the equation by 21. Again, it's crucial to perform the same operation on both sides to maintain the equation's balance.

Dividing both sides by 21, we get: (21x) / 21 = -63 / 21. On the left side, 21 divided by 21 is 1, leaving us with 1x, which is simply written as 'x'. On the right side, -63 divided by 21 is -3. Therefore, our equation simplifies to: x = -3. We have successfully solved for 'x'! This result tells us that the value of 'x' that makes the original equation true is -3.

This final step underscores the power of inverse operations in solving equations. By dividing both sides by the coefficient of 'x', we effectively "undid" the multiplication and revealed the value of 'x'. Solving for 'x' is the culmination of all the previous steps, and it provides the answer we were seeking. The solution, x = -3, is not just a number; it's the value that satisfies the original equation. To be absolutely sure, we could substitute -3 back into the original equation to verify that it holds true. This process of solving for 'x' demonstrates the elegance and precision of algebraic manipulation. By following these step-by-step procedures, we can confidently tackle a wide range of linear equations.

Step 4: Check Your Answer (Optional but Recommended)

While we've confidently solved for 'x' and arrived at the solution x = -3, it's always a good practice to check our answer. This step, though optional, is highly recommended, especially in academic settings or when accuracy is paramount. Checking our answer involves substituting the value we found for 'x' back into the original equation. If the equation holds true after the substitution, we can be confident in our solution. If not, it indicates a potential error in our steps, prompting us to review our work.

Our original equation was 16x - 4 + 5x = -67. To check our solution, we replace 'x' with -3: 16(-3) - 4 + 5(-3) = -67. Now, we perform the arithmetic operations. 16 multiplied by -3 is -48. 5 multiplied by -3 is -15. So, the equation becomes: -48 - 4 - 15 = -67. Next, we combine the terms on the left side: -48 minus 4 is -52, and -52 minus 15 is -67. Therefore, we have -67 = -67. This confirms that our solution, x = -3, is correct. The left side of the equation equals the right side, validating our answer. Checking our answer not only provides assurance but also reinforces our understanding of the equation and the solution process. It's a valuable habit to cultivate in your mathematical journey, helping to prevent errors and build confidence in your problem-solving abilities.

Conclusion: Mastering Linear Equations

In this comprehensive guide, we've meticulously walked through the process of solving the linear equation 16x - 4 + 5x = -67. From combining like terms to isolating the variable and finally solving for 'x', we've dissected each step, emphasizing the underlying principles and techniques. We've also highlighted the importance of checking our answer to ensure accuracy. The solution we arrived at, x = -3, is not just a numerical value; it's the key that unlocks the equation, making it a true statement.

The ability to solve linear equations is a cornerstone of algebra and a fundamental skill in mathematics. It's a skill that transcends the classroom, finding applications in various fields, from science and engineering to economics and everyday problem-solving. By mastering the techniques discussed in this article, you're equipping yourself with a powerful tool for tackling a wide range of mathematical challenges. Remember, the key to success in algebra lies in understanding the core principles and practicing consistently. Each equation you solve, each problem you tackle, strengthens your understanding and builds your confidence. So, embrace the challenge, delve into the world of linear equations, and unlock your mathematical potential. With practice and perseverance, you'll find that solving equations becomes not just a task, but a rewarding intellectual endeavor.