Solving The Equation (x/-4)-5=4 A Step-by-Step Guide

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In this article, we will delve into the process of solving the equation x−4−5=4{\frac{x}{-4}-5=4}. This is a fundamental algebraic problem that requires a systematic approach to isolate the variable x. Understanding how to solve such equations is crucial for mastering more complex mathematical concepts. We will break down each step, providing clear explanations and justifications to ensure a comprehensive understanding. This step-by-step guide aims to equip you with the skills necessary to confidently tackle similar algebraic problems.

Understanding the Basics of Algebraic Equations

Before we dive into solving the specific equation, it's essential to grasp the basic principles of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can contain numbers, variables, and mathematical operations. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. This involves isolating the variable on one side of the equation by performing the same operations on both sides, maintaining the balance of the equation. The fundamental principle behind solving algebraic equations is the concept of inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, as are multiplication and division. By applying inverse operations, we can systematically eliminate terms from one side of the equation until the variable is isolated.

Key concepts in algebra include variables, constants, coefficients, and operators. A variable is a symbol (usually a letter) that represents an unknown value. A constant is a fixed numerical value. A coefficient is a number that multiplies a variable. Operators are symbols that indicate mathematical operations, such as addition, subtraction, multiplication, and division. When solving equations, it's crucial to follow the order of operations (PEMDAS/BODMAS) in reverse to correctly isolate the variable. This means addressing addition and subtraction before multiplication and division. Understanding these basic principles is essential for successfully navigating algebraic equations and finding solutions. The ability to manipulate equations and isolate variables is a foundational skill in mathematics and is applicable in various fields, including science, engineering, and economics. Mastery of these concepts opens the door to more advanced mathematical topics and real-world problem-solving.

Step 1: Isolating the Term with the Variable

To begin solving the equation x−4−5=4{\frac{x}{-4}-5=4}, our first objective is to isolate the term that contains the variable x. In this case, the term is x−4{\frac{x}{-4}}. To isolate this term, we need to eliminate the constant term that is being subtracted from it, which is -5. The principle behind isolating the variable term is to use the inverse operation. Since 5 is being subtracted from the term x−4{\frac{x}{-4}}, we need to add 5 to both sides of the equation. This maintains the equality of the equation and allows us to simplify it.

Adding 5 to both sides of the equation, we get:

x−4−5+5=4+5{\frac{x}{-4}-5+5=4+5}

This simplifies to:

x−4=9{\frac{x}{-4}=9}

By adding 5 to both sides, we have successfully isolated the term x−4{\frac{x}{-4}} on the left side of the equation. This is a crucial step in solving for x because it brings us closer to isolating the variable itself. Isolating the variable term is a fundamental technique in algebra, allowing us to gradually simplify the equation and make it easier to solve. This step demonstrates the importance of using inverse operations to manipulate equations and maintain balance. The ability to isolate terms is a key skill in algebra, paving the way for solving more complex equations. It is a foundational concept that is applied repeatedly in various mathematical contexts.

Step 2: Solving for x

Now that we have isolated the term x−4{\frac{x}{-4}} and the equation is simplified to x−4=9{\frac{x}{-4}=9}, the next step is to solve for x. To do this, we need to eliminate the denominator -4 from the left side of the equation. The operation currently being applied to x is division by -4. To undo this division, we need to perform the inverse operation, which is multiplication. We will multiply both sides of the equation by -4. This ensures that the equality of the equation is maintained while allowing us to isolate x.

Multiplying both sides of the equation by -4, we get:

x−4×−4=9×−4{\frac{x}{-4} \times -4 = 9 \times -4}

On the left side, the multiplication by -4 cancels out the division by -4, leaving us with x. On the right side, we multiply 9 by -4, which gives us -36. Therefore, the equation simplifies to:

x=−36{x = -36}

Thus, we have successfully solved for x, and the solution to the equation x−4−5=4{\frac{x}{-4}-5=4} is x = -36. This step demonstrates the power of using inverse operations to isolate the variable and solve for its value. Multiplying both sides of the equation by -4 effectively eliminated the denominator, allowing us to directly determine the value of x. This process highlights the systematic approach to solving algebraic equations, where each step builds upon the previous one to gradually simplify the equation and reveal the solution. The ability to confidently manipulate equations and isolate variables is a critical skill in mathematics, and this example provides a clear illustration of how to apply this skill.

Step 3: Verifying the Solution

After solving an equation, it's crucial to verify the solution to ensure its correctness. This step helps to catch any potential errors made during the solving process and provides confidence in the answer. To verify the solution, we substitute the value we found for x back into the original equation. If the solution is correct, the left side of the equation should equal the right side when x is replaced with its value. In our case, we found that x = -36. Let's substitute this value into the original equation x−4−5=4{\frac{x}{-4}-5=4}:

−36−4−5=4{\frac{-36}{-4}-5=4}

First, we simplify the fraction −36−4{\frac{-36}{-4}}. Dividing -36 by -4 gives us 9. So, the equation becomes:

9−5=4{9-5=4}

Next, we perform the subtraction on the left side: 9 - 5 = 4. This simplifies the equation to:

4=4{4=4}

Since the left side of the equation equals the right side, our solution x = -36 is correct. Verifying the solution is an essential step in the problem-solving process. It confirms that our calculations and algebraic manipulations were accurate and that the value we found for x indeed satisfies the original equation. This step reinforces the understanding of equation solving and helps to develop a habit of checking work for accuracy. The process of substitution and simplification is a powerful tool for verifying solutions and building confidence in one's mathematical abilities. It is a practice that should be applied to all equation-solving problems to ensure the correctness of the final answer.

Conclusion

In conclusion, we have successfully solved the equation x−4−5=4{\frac{x}{-4}-5=4} by following a systematic step-by-step approach. We began by isolating the term with the variable, which involved adding 5 to both sides of the equation. This simplified the equation to x−4=9{\frac{x}{-4}=9}. Next, we solved for x by multiplying both sides of the equation by -4, resulting in x = -36. Finally, we verified our solution by substituting x = -36 back into the original equation, confirming that it is indeed the correct solution.

This process highlights the fundamental principles of solving algebraic equations, which include using inverse operations to isolate variables and maintaining the balance of the equation by performing the same operations on both sides. The ability to solve equations like this is a crucial skill in mathematics, forming the basis for more advanced topics. Understanding the steps involved and the reasoning behind them is essential for building a strong foundation in algebra. By mastering these techniques, one can confidently tackle a wide range of mathematical problems. The systematic approach demonstrated in this article provides a clear framework for solving similar equations and reinforces the importance of accuracy and verification in mathematical problem-solving. The skills acquired through solving this equation are transferable to various mathematical and real-world scenarios, making it a valuable learning experience.