Solving The Equation 5 = X/4 - 4 A Step-by-Step Guide

In this article, we'll dive deep into the process of solving linear equations, focusing on the specific equation 5 = x/4 - 4. This type of equation is fundamental in algebra, and mastering the steps to solve it will provide you with a solid foundation for tackling more complex mathematical problems. We'll break down each step, explaining the reasoning behind it, and ultimately arrive at the solution for x. Understanding how to isolate variables and manipulate equations is a crucial skill in mathematics and many other fields, making this a valuable concept to grasp. This equation, 5 = x/4 - 4, exemplifies a classic linear equation structure, where our goal is to determine the value of the unknown variable, 'x'. To do this effectively, we'll employ the principles of inverse operations, carefully undoing the operations performed on 'x' to isolate it on one side of the equation. This methodical approach ensures accuracy and builds a deeper understanding of algebraic problem-solving. The journey to solving for x involves a series of deliberate steps, each designed to bring us closer to the solution. We begin by identifying the operations affecting x and then systematically reverse them, maintaining the balance of the equation. This principle of maintaining balance is paramount in algebra, as it ensures that any operation performed on one side of the equation is mirrored on the other, preserving the equality and leading us to the correct answer. By understanding and applying these steps, you will not only solve this particular equation but also gain the confidence to tackle a wide range of linear equations.

Understanding the Basics of Linear Equations

Before we jump into solving our equation, let's briefly touch on the basics of linear equations. A linear equation is an equation that can be written in the form ax + b = c, where a, b, and c are constants, and x is the variable. The key characteristic of a linear equation is that the variable is raised to the power of 1. Our equation, 5 = x/4 - 4, fits this form, although it might not be immediately obvious. We can rewrite x/4 as (1/4)x, making it clearer that the equation is indeed linear. Understanding this foundational form helps us recognize and approach linear equations with a consistent strategy. The goal in solving any linear equation is to isolate the variable, which means getting x by itself on one side of the equation. This isolation is achieved through a series of inverse operations, which are operations that undo each other. For instance, addition and subtraction are inverse operations, as are multiplication and division. By applying these inverse operations strategically, we can peel away the constants surrounding x until it stands alone, revealing its value. Linear equations are the building blocks of more advanced mathematical concepts, making a solid understanding of their solutions essential for further studies in algebra and beyond. Mastering the techniques for solving linear equations not only empowers you to tackle mathematical challenges but also hones your problem-solving skills, which are valuable in various aspects of life.

Step 1: Isolate the term with x

Our first step in solving 5 = x/4 - 4 is to isolate the term containing x, which is x/4. To do this, we need to eliminate the constant term that's being subtracted from it, which is -4. The inverse operation of subtraction is addition, so we'll add 4 to both sides of the equation. This maintains the balance of the equation, ensuring that we're performing the same operation on both sides. By adding 4 to both sides, we effectively cancel out the -4 on the right side, leaving us with just the x/4 term. This is a crucial step in isolating x and bringing us closer to the solution. The equation now becomes 5 + 4 = x/4 - 4 + 4, which simplifies to 9 = x/4. This step highlights the importance of inverse operations in solving equations. By strategically applying the inverse of subtraction (addition), we've successfully isolated the term containing our variable, setting the stage for the next step in our solution process. This methodical approach is key to solving any algebraic equation, ensuring accuracy and clarity in our work. Remember, the goal is to undo the operations affecting x, and in this case, we've successfully undone the subtraction, paving the way for isolating x completely.

Step 2: Eliminate the denominator

Now that we have 9 = x/4, the next step is to eliminate the denominator, which is 4. The x term is being divided by 4, so the inverse operation we need to apply is multiplication. We'll multiply both sides of the equation by 4. This will cancel out the division by 4 on the right side, leaving us with just x. Remember, whatever we do to one side of the equation, we must do to the other to maintain the balance and ensure the equation remains true. Multiplying both sides by 4 gives us 4 * 9 = 4 * (x/4). On the left side, 4 * 9 equals 36. On the right side, the 4 in the numerator and the 4 in the denominator cancel each other out, leaving us with just x. This simplifies our equation to 36 = x. This step demonstrates the power of using inverse operations to isolate variables. By multiplying both sides of the equation by the denominator, we effectively undid the division, bringing us one step closer to the solution. This principle applies to any equation with a fractional term; multiplying by the denominator is a key technique for simplifying the equation and isolating the variable. The equation 36 = x is almost our final answer, as x is now isolated. We've successfully eliminated both the constant term and the denominator, paving the way for the final solution.

Step 3: State the solution

After performing the previous steps, we've arrived at the equation 36 = x. This equation clearly states the solution for x: x equals 36. Therefore, the solution to the equation 5 = x/4 - 4 is x = 36. We can also write this as x = 36, which is the standard way of expressing the solution. This final step is crucial, as it explicitly states the value of x that satisfies the original equation. It's a clear and concise answer that leaves no room for ambiguity. To ensure our solution is correct, we can substitute x = 36 back into the original equation and check if it holds true. Substituting 36 for x in 5 = x/4 - 4 gives us 5 = 36/4 - 4. Simplifying the right side, we get 5 = 9 - 4, which further simplifies to 5 = 5. This confirms that our solution, x = 36, is indeed correct. Stating the solution clearly and verifying it by substituting it back into the original equation are important steps in the problem-solving process. They ensure accuracy and provide confidence in the answer. With this final step, we've successfully solved the equation and found the value of x.

Verifying the Solution

To be absolutely certain that our solution x = 36 is correct, it's essential to verify it. We do this by substituting the value of x back into the original equation, 5 = x/4 - 4. If the equation holds true after the substitution, then we know our solution is accurate. Substituting x = 36 into the equation gives us 5 = 36/4 - 4. Now, we simplify the right side of the equation. First, we divide 36 by 4, which equals 9. So, the equation becomes 5 = 9 - 4. Next, we subtract 4 from 9, which equals 5. This gives us 5 = 5. Since both sides of the equation are equal, we have successfully verified our solution. This verification step is a crucial part of the problem-solving process. It provides a check against any potential errors made during the algebraic manipulations. By substituting the solution back into the original equation, we can confirm its validity and gain confidence in our answer. Verifying the solution not only ensures accuracy but also reinforces the understanding of the equation and the steps taken to solve it. It's a valuable habit to develop when solving any mathematical problem.

Conclusion: Mastering Linear Equations

In conclusion, we've successfully solved the equation 5 = x/4 - 4 by following a step-by-step approach. We isolated the term with x, eliminated the denominator, and arrived at the solution x = 36. We also verified our solution by substituting it back into the original equation. This process demonstrates the fundamental principles of solving linear equations, which involve using inverse operations to isolate the variable. Mastering these principles is crucial for success in algebra and beyond. Linear equations are the foundation upon which many other mathematical concepts are built. By understanding how to solve them efficiently and accurately, you'll be well-equipped to tackle more complex problems. The key takeaways from this exercise are the importance of inverse operations, maintaining balance in equations, and verifying solutions. These skills are not only valuable in mathematics but also in various other fields that require problem-solving and analytical thinking. Practice is essential for mastering linear equations. The more you practice, the more comfortable and confident you'll become in solving them. With consistent effort and a solid understanding of the principles involved, you can conquer any linear equation that comes your way. Remember, mathematics is a skill that improves with practice, and linear equations are a great starting point for building your mathematical foundation.

Therefore, the correct answer is A. x=36