Step-by-Step Solution Of The Equation X/4 + 5 = 23
Let's delve into the step-by-step solution.
Step 1: The Initial Equation
Our journey begins with the equation itself. The initial equation sets the stage for our problem-solving adventure. Understanding what the equation represents is paramount. In this case, we have a variable, 'x', which is divided by 4, and then 5 is added to the result, ultimately equaling 23. This equation is a linear equation, a type of equation commonly encountered in algebra. Linear equations involve variables raised to the power of 1, and they typically have one solution. Our goal is to isolate 'x' on one side of the equation to determine its value. To achieve this, we will employ a series of algebraic manipulations, ensuring that we maintain the balance of the equation at all times. The principle of maintaining balance is a cornerstone of equation solving – what we do to one side, we must also do to the other. This ensures that the equality remains valid throughout the process. The initial equation, $\frac{x}{4}+5=23$, is the foundation upon which our solution will be built. It represents a specific relationship between 'x' and the numerical values in the equation. Before we proceed to the next step, it's crucial to have a clear grasp of the equation's structure and what we aim to accomplish. We are essentially trying to 'undo' the operations performed on 'x' to reveal its true value. Think of it as peeling away layers to get to the core. Each step we take will bring us closer to isolating 'x' and finding the solution. By carefully examining the initial equation, we can strategize our approach and anticipate the necessary steps. This initial analysis is key to effective problem-solving in mathematics. Remember, a strong understanding of the starting point is essential for navigating the solution process successfully. We need to isolate x which the core of what we are looking for.
Step 2: Isolating the Term with the Variable
The next crucial step in solving equations is to isolate the term containing the variable. In our equation, $\frac{x}{4}+5=23$, the term with the variable 'x' is $\frac{x}{4}$. To isolate this term, we need to eliminate the constant term, which is +5, from the left side of the equation. The fundamental principle we employ here is the inverse operation. The inverse operation of addition is subtraction, so we subtract 5 from both sides of the equation. This ensures that the equation remains balanced. By subtracting 5 from both sides, we effectively 'undo' the addition of 5 on the left side, bringing us closer to isolating the term with 'x'. The step can be represented as follows:
This step is a critical juncture in the solution process. It demonstrates our understanding of how to manipulate equations while preserving their integrity. The balance of the equation is paramount, and we maintain this balance by performing the same operation on both sides. Subtracting 5 from both sides is a strategic move, designed to simplify the equation and pave the way for further steps. It's like removing a roadblock in our journey to finding the value of 'x'. By isolating the term with the variable, we are setting the stage for the next operation, which will involve eliminating the division by 4. Each step in the equation-solving process is carefully chosen to bring us closer to the solution, and this step is no exception. It's a testament to the power of algebraic manipulation and the importance of understanding inverse operations. Remember, the goal is to isolate 'x', and this step is a significant stride in that direction. Now, with the constant term eliminated, we can focus our attention on dealing with the division by 4. This is a systematic and logical approach to solving equations, ensuring that we arrive at the correct solution.
Step 3: Simplifying the Equation
After subtracting 5 from both sides of the equation in the previous step, we now have a simplified equation. Simplifying equations is a crucial step in the problem-solving process as it makes the equation easier to work with and understand. The equation we arrived at in Step 2 is:
This simplified form is obtained by performing the subtraction on both sides of the equation: 5 - 5 cancels out on the left side, and 23 - 5 equals 18 on the right side. This simplification is not just about making the equation look neater; it's about revealing the core relationship between the variable 'x' and the numerical values. By removing the unnecessary clutter, we can clearly see that 'x' divided by 4 equals 18. This clearer view allows us to strategize the next step more effectively. The simplified equation, $\frac{x}{4} = 18$, is a significant milestone in our journey to solving for 'x'. It represents a direct relationship between 'x' and the number 18, with the only remaining operation being the division by 4. This means we are very close to isolating 'x' and finding its value. The process of simplification is a fundamental aspect of algebra. It involves applying arithmetic operations and algebraic rules to reduce an equation to its simplest form. This not only makes the equation easier to solve but also helps in understanding the underlying mathematical relationships. In this case, simplification has allowed us to clearly see the next step required to isolate 'x'. Think of simplification as a process of refinement, where we remove the extraneous elements to reveal the essential components. This makes the problem more manageable and the solution more attainable. With the equation now in its simplified form, we are well-positioned to proceed to the final steps and determine the value of 'x'. The clarity gained through simplification is invaluable in solving equations efficiently and accurately. Remember, a simplified equation is a powerful tool in mathematical problem-solving.
Step 4: Isolating the Variable by Multiplication
Now that we have the simplified equation $\frac{x}{4} = 18$, our next objective is to completely isolate the variable 'x'. Currently, 'x' is being divided 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 equation remains balanced, a principle that is paramount in solving equations. Multiplying both sides by 4 will effectively cancel out the division by 4 on the left side, leaving 'x' isolated. This step is represented as:
This step is a crucial application of the multiplication property of equality, which states that if we multiply both sides of an equation by the same non-zero number, the equation remains balanced. By multiplying by 4, we are strategically eliminating the denominator and bringing us closer to the solution. This is a common technique used in algebra to isolate variables and solve equations. The act of multiplying both sides by 4 is like using a key to unlock the value of 'x'. It reverses the division operation and reveals the true value of the variable. Each step in the equation-solving process is carefully designed to build upon the previous step, and this step is no exception. It's a testament to the power of inverse operations and the importance of maintaining balance in equations. The result of this multiplication will give us the value of 'x'. This step demonstrates a clear understanding of how to manipulate equations to achieve a specific goal. The goal, in this case, is to isolate 'x', and multiplication is the tool we are using to achieve that goal. With 'x' now isolated, the final step will be to perform the multiplication on the right side of the equation to determine the numerical value of 'x'. Remember, isolating the variable is the ultimate aim in solving equations, and this step brings us one step closer to that aim.
Step 5: The Solution
Having multiplied both sides of the equation by 4 in the previous step, we now arrive at the solution. The solution to an equation is the value of the variable that makes the equation true. Let's recap the equation from Step 4:
On the left side, the multiplication by 4 cancels out the division by 4, leaving us with 'x'. On the right side, we need to perform the multiplication: 4 multiplied by 18. This gives us 72. Therefore, the equation simplifies to:
This is the final solution to the equation. It tells us that the value of 'x' that satisfies the original equation $\frac{x}{4}+5=23$ is 72. The solution represents the culmination of all the steps we have taken. It's the answer we were seeking, and it's the result of careful and systematic algebraic manipulation. Finding the solution is the ultimate goal in solving equations. It's the point at which we have successfully isolated the variable and determined its value. The solution, x = 72, is not just a number; it's a piece of information that completes the equation. It tells us the specific value that, when substituted for 'x' in the original equation, will make the equation true. This is the essence of what it means to solve an equation. The process of arriving at the solution has involved a series of strategic steps, each designed to bring us closer to the answer. From isolating the term with the variable to simplifying the equation and using inverse operations, each step has played a crucial role. Now that we have found the solution, it's always a good practice to check our answer by substituting it back into the original equation. This confirms that our solution is correct and that we have not made any errors in the process. Remember, the solution is the key that unlocks the equation, revealing the value of the unknown variable.
Conclusion
In conclusion, solving the equation $\frac{x}{4}+5=23$ involves a series of logical steps, each building upon the previous one. We began by understanding the initial equation, then systematically isolated the variable 'x' by using inverse operations and maintaining the balance of the equation. The steps included subtracting 5 from both sides, simplifying the equation, multiplying both sides by 4, and finally arriving at the solution: x = 72. Mastering the art of solving equations is a fundamental skill in mathematics. It requires a clear understanding of algebraic principles, a systematic approach, and attention to detail. By following these steps, we can confidently solve a wide range of linear equations. This step-by-step guide serves as a valuable resource for anyone seeking to enhance their equation-solving abilities. Remember, practice is key to mastery, so continue to solve equations and refine your skills. The ability to solve equations is not only essential for academic success but also for problem-solving in various real-world scenarios. From calculating finances to designing structures, equations are the foundation of many practical applications. The process of solving equations is a journey of discovery, where we unravel the relationship between variables and constants to find the hidden value. It's a process that requires patience, persistence, and a willingness to learn from mistakes. By embracing the challenge and applying the principles outlined in this guide, you can become a proficient equation solver. This skill will serve you well in your mathematical endeavors and beyond. The power to solve equations is the power to unlock solutions to a myriad of problems. So, continue to explore the world of mathematics and embrace the challenges that come your way. With each equation you solve, you are building a stronger foundation for future success. Remember, the journey of a thousand miles begins with a single step, and the journey of mastering equations begins with understanding the fundamentals.
