Solving 16x - 4 + 5x = -67 Step-by-Step Guide
In the realm of mathematics, particularly in algebra, solving for unknown variables is a fundamental skill. This article will delve into the process of solving a linear equation, specifically the equation 16x - 4 + 5x = -67. We will break down the steps involved, providing a clear and concise explanation for each, ensuring a solid understanding of how to arrive at the correct solution. This guide aims to not only provide the answer but also to equip you with the knowledge and confidence to tackle similar algebraic problems. Understanding the principles behind solving equations is crucial for various applications in mathematics and other scientific fields. Let's embark on this journey of algebraic exploration and unlock the value of x!
Understanding the Equation
The equation we aim to solve is 16x - 4 + 5x = -67. This is a linear equation, meaning the highest power of the variable x is 1. To solve for x, our goal is to isolate x on one side of the equation. This involves using algebraic manipulations to combine like terms and move constants to the opposite side. The key principle we'll employ is maintaining the equality of the equation. Whatever operation we perform on one side, we must also perform on the other side. This ensures that the balance of the equation is preserved, leading us to the accurate value of x. Before we dive into the steps, it's essential to grasp the structure of the equation and the role of each component. The terms involving x (16x and 5x) are variable terms, while the constants (-4 and -67) are fixed numerical values. Our strategy will involve combining the x terms, moving the constant term to the right side, and then isolating x. With a clear understanding of these fundamental concepts, we can confidently proceed to solve the equation.
Step-by-Step Solution
To effectively solve the equation 16x - 4 + 5x = -67, we'll follow a series of logical steps, ensuring clarity and accuracy in our calculations. Each step is crucial in isolating the variable x and ultimately finding its value.
Step 1: Combine Like Terms
The first step involves simplifying the equation by combining like terms on the left side. We have two terms involving x: 16x and 5x. Adding these together, we get:
16x + 5x = 21x
So, the equation now becomes:
21x - 4 = -67
This step consolidates the x terms, making the equation more manageable. It's a fundamental algebraic technique that simplifies expressions and lays the groundwork for further steps.
Step 2: Isolate the Variable Term
Next, we want to isolate the term with x (21x) on one side of the equation. To do this, we need to eliminate the constant term (-4) from the left side. We achieve this by adding 4 to both sides of the equation:
21x - 4 + 4 = -67 + 4
This simplifies to:
21x = -63
By adding 4 to both sides, we maintain the equation's balance while moving the constant term to the right side, bringing us closer to isolating x.
Step 3: Solve for x
Finally, to solve for x, we need to isolate it completely. Currently, x is multiplied by 21. To undo this multiplication, we divide both sides of the equation by 21:
21x / 21 = -63 / 21
This gives us:
x = -3
Therefore, the solution for x in the equation 16x - 4 + 5x = -67 is -3. This completes the step-by-step solution process, demonstrating how algebraic manipulation can lead us to the value of the unknown variable.
Verifying the Solution
In mathematics, it's always a good practice to verify the solution obtained to ensure its accuracy. This involves substituting the calculated value of x back into the original equation and checking if both sides of the equation are equal. In our case, we found that x = -3. Let's substitute this value into the original equation:
16x - 4 + 5x = -67
Substituting x = -3, we get:
16(-3) - 4 + 5(-3) = -67
Now, let's simplify the left side of the equation:
-48 - 4 - 15 = -67
-67 = -67
As we can see, both sides of the equation are equal, confirming that our solution x = -3 is indeed correct. This verification step reinforces our confidence in the accuracy of our calculations and demonstrates the importance of checking our work in mathematical problem-solving. By substituting the solution back into the original equation, we ensure that it satisfies the given condition, leaving no room for doubt about the correctness of our answer.
Common Mistakes to Avoid
When solving algebraic equations, certain common mistakes can lead to incorrect answers. Being aware of these pitfalls can significantly improve accuracy and problem-solving skills. One frequent error is incorrectly combining like terms. For instance, failing to recognize the signs of the terms or adding unlike terms together. In our equation, 16x - 4 + 5x = -67, a mistake could be made by incorrectly adding 16x and 5x or by improperly handling the constant term -4. Another common mistake is not applying the same operation to both sides of the equation. To maintain equality, any operation performed on one side must be mirrored on the other side. For example, when adding 4 to both sides, if it's only added to the left side, the equation becomes unbalanced, leading to an incorrect solution. Sign errors are also prevalent, particularly when dealing with negative numbers. For example, mishandling the negative sign in -67 or during the division step can result in an incorrect value for x. Lastly, neglecting the order of operations (PEMDAS/BODMAS) can lead to calculation errors. It's crucial to perform operations in the correct sequence to arrive at the accurate solution. By being mindful of these common mistakes and practicing careful, methodical problem-solving, you can minimize errors and enhance your algebraic proficiency.
Alternative Approaches
While the step-by-step method outlined earlier is a standard approach to solving linear equations, exploring alternative methods can provide a deeper understanding and enhance problem-solving flexibility. One alternative approach involves rearranging the equation differently before combining like terms. For instance, instead of directly combining 16x and 5x, one could first add 4 to both sides of the equation, resulting in 16x + 5x = -63. This rearrangement doesn't change the solution but offers a different perspective on the problem. Another technique involves visualizing the equation using a balance scale analogy. Imagine the equation as a scale where both sides must remain balanced. Any operation performed on one side must be mirrored on the other to maintain equilibrium. This visual representation can aid in understanding the concept of equality and the importance of performing operations on both sides. Furthermore, using online equation solvers or calculators can serve as a tool for verifying solutions and exploring different solution paths. These tools can provide step-by-step solutions, highlighting alternative approaches and potential shortcuts. While it's essential to develop manual problem-solving skills, utilizing such resources can enhance learning and provide a deeper understanding of algebraic concepts. By exploring alternative approaches, learners can develop a more robust understanding of equation-solving techniques and improve their mathematical intuition.
Real-World Applications
The ability to solve linear equations, like the one we addressed (16x - 4 + 5x = -67), extends far beyond the classroom and finds numerous applications in real-world scenarios. These equations are fundamental tools in various fields, including science, engineering, finance, and economics. In physics, linear equations are used to model motion, forces, and energy. For example, calculating the distance traveled by an object at a constant speed involves solving a linear equation. In engineering, these equations are crucial for designing structures, circuits, and systems. Determining the appropriate dimensions and specifications often requires solving linear equations that represent the relationships between different variables. Financial planning heavily relies on linear equations for budgeting, calculating interest, and determining loan payments. Understanding how to solve these equations is essential for making informed financial decisions. Economics uses linear equations to model supply and demand, analyze market trends, and predict economic outcomes. Businesses use these equations for cost analysis, pricing strategies, and forecasting profits. The ability to solve linear equations is a valuable skill in everyday life as well. From calculating discounts and sales tax to determining the cost of a service or splitting a bill among friends, linear equations provide a framework for solving practical problems. The mastery of these equations empowers individuals to make informed decisions, solve problems effectively, and navigate various aspects of daily life with greater confidence. Therefore, understanding the principles behind solving equations like 16x - 4 + 5x = -67 is not just an academic exercise but a valuable life skill.
Conclusion
In conclusion, solving the equation 16x - 4 + 5x = -67 provides a clear illustration of the fundamental principles of algebra. By combining like terms, isolating the variable term, and applying the appropriate operations, we successfully determined that x = -3. The step-by-step solution not only provides the answer but also reinforces the importance of each stage in the algebraic process. Verifying the solution by substituting it back into the original equation further solidifies our confidence in the result. Understanding the common mistakes to avoid, such as sign errors and incorrect order of operations, helps to minimize errors and improve accuracy in problem-solving. Exploring alternative approaches to solving equations broadens our understanding and enhances flexibility in tackling different types of problems. The real-world applications of linear equations highlight the practical relevance of algebraic skills in various fields and everyday situations. Mastering the techniques for solving linear equations, like the one discussed, equips us with valuable tools for problem-solving, critical thinking, and decision-making. The ability to confidently manipulate equations and solve for unknown variables is a cornerstone of mathematical literacy and a skill that benefits us in numerous aspects of life. Therefore, the process of solving 16x - 4 + 5x = -67 serves as a valuable learning experience, reinforcing algebraic principles and empowering us to tackle future mathematical challenges with greater proficiency.
