Solving 10 = 2 - 4(ax - 3) Step-by-Step Guide
Linear equations are the foundation of algebra, and mastering the techniques to solve them is crucial for success in mathematics and various other fields. These equations, characterized by a variable raised to the power of one, appear in numerous real-world applications, from calculating finances to designing structures. In this comprehensive guide, we will delve into the process of solving a specific linear equation, providing a clear and methodical approach that can be applied to a wide range of similar problems. By understanding the underlying principles and practicing the steps involved, you can confidently tackle any linear equation that comes your way. Solving linear equations isn't just about finding the answer; it's about developing a systematic approach to problem-solving that will serve you well in many different contexts.
The equation we will be focusing on is:
Our goal is to isolate the variable 'x' on one side of the equation, thereby determining its value. This involves a series of algebraic manipulations, each carefully designed to maintain the equation's balance. Let's embark on this journey together, breaking down each step and highlighting the reasoning behind it. We'll cover the fundamentals, such as the order of operations, the distributive property, and the concept of inverse operations, to ensure a thorough understanding. Remember, the key to success in algebra is practice, so we encourage you to follow along, work through the examples, and try solving similar equations on your own. By the end of this guide, you'll have the tools and knowledge to confidently solve linear equations and apply them to real-world scenarios.
Step 1: Distribute the -4
The first step in solving this equation is to simplify the expression by distributing the -4 across the terms inside the parentheses. The distributive property states that a(b + c) = ab + ac. Applying this property to our equation, we get:
10 = 2 - 4(ax - 3)
10 = 2 - 4ax + 12
Explanation:
We multiply -4 by both 'ax' and '-3'. Remember that multiplying a negative number by a negative number results in a positive number. So, -4 multiplied by -3 equals +12. This step effectively removes the parentheses and allows us to combine like terms in the next step. Understanding the distributive property is fundamental in algebra, as it allows us to simplify complex expressions and make them easier to work with. The key is to ensure that each term inside the parentheses is multiplied by the factor outside. This process is crucial for isolating the variable and ultimately solving the equation. Mastering this step will significantly improve your ability to handle algebraic equations efficiently.
Step 2: Combine Like Terms
Now that we've distributed the -4, we can simplify the right side of the equation by combining the constant terms, 2 and 12. Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. In this case, 2 and 12 are both constants, so we can simply add them together:
10 = 2 - 4ax + 12
10 = 14 - 4ax
Explanation:
By adding 2 and 12, we reduce the number of terms in the equation, making it easier to isolate the variable 'x'. This step is crucial for streamlining the equation and moving closer to the solution. Identifying and combining like terms is a fundamental skill in algebra. It simplifies the equation, allowing us to focus on isolating the variable. This process involves recognizing terms that share the same variable and exponent, or, in the case of constants, simply adding or subtracting their values. Mastering this technique is essential for efficiently solving equations and tackling more complex algebraic problems.
Step 3: Isolate the Term with x
To isolate the term containing 'x' (-4ax), we need to get rid of the constant term (14) on the right side of the equation. We can do this by subtracting 14 from both sides of the equation. Remember, to maintain the equality, any operation performed on one side of the equation must also be performed on the other side. This principle is the cornerstone of algebraic manipulation.
10 = 14 - 4ax
10 - 14 = 14 - 4ax - 14
-4 = -4ax
Explanation:
Subtracting 14 from both sides cancels out the +14 on the right side, leaving us with -4ax. This step brings us closer to isolating 'x'. The concept of maintaining balance in an equation is crucial in algebra. By performing the same operation on both sides, we ensure that the equality remains valid. This principle allows us to manipulate equations strategically, isolating the variable we want to solve for. In this case, subtracting 14 from both sides is a key step in isolating the term containing 'x', paving the way for the final step of solving for 'x' itself.
Step 4: Solve for x
Finally, to solve for 'x', we need to isolate it by dividing both sides of the equation by -4a. This will undo the multiplication of 'x' by -4a.
-4 = -4ax
-4 / (-4a) = -4ax / (-4a)
1/a = x
Explanation:
Dividing both sides by -4a cancels out the -4a on the right side, leaving us with 'x' isolated. On the left side, -4 divided by -4a simplifies to 1/a. Therefore, the solution to the equation is x = 1/a.
Thus, the solution to the equation 10 = 2 - 4(ax - 3) is x = 1/a, which corresponds to option A.
Conclusion
In this comprehensive guide, we have meticulously walked through the steps to solve the linear equation 10 = 2 - 4(ax - 3) for x. We began by distributing the -4, then combined like terms, isolated the term containing x, and finally, solved for x by dividing both sides of the equation by the coefficient of x. The final answer, x = 1/a, demonstrates the power of algebraic manipulation in isolating and determining the value of a variable. Throughout the process, we emphasized the importance of maintaining balance in the equation by performing the same operations on both sides.
Mastering the techniques presented in this guide is crucial for success in algebra and beyond. Linear equations are ubiquitous in mathematics and its applications, appearing in various contexts, from physics and engineering to economics and finance. The ability to confidently solve these equations is a valuable skill that will serve you well in numerous fields. We encourage you to practice these steps with other linear equations to solidify your understanding and develop your problem-solving abilities. Remember, the key to success in mathematics is consistent effort and a willingness to learn from your mistakes. By diligently practicing and applying the principles outlined in this guide, you can confidently tackle any linear equation that comes your way.
Furthermore, understanding the underlying principles of solving linear equations lays the foundation for more advanced mathematical concepts. As you progress in your mathematical journey, you will encounter more complex equations and systems of equations. The skills and techniques you develop while solving linear equations will be invaluable in tackling these challenges. For instance, the concept of inverse operations, which we used to isolate x, is fundamental in solving various types of equations. Similarly, the principle of maintaining balance in an equation is a cornerstone of algebraic manipulation and applies to a wide range of mathematical problems. Therefore, mastering linear equations is not just about finding the solution to a specific problem; it's about developing a solid foundation for future mathematical success. We encourage you to explore further resources and seek out opportunities to apply these skills in different contexts. The more you practice and engage with these concepts, the more confident and proficient you will become in your mathematical abilities.
