Solving For The Unknown If Four Times A Number Plus 3 Equals 11
Introduction
In the realm of mathematics, we often encounter intriguing puzzles that challenge our problem-solving skills. One such puzzle involves the interplay of numbers and operations, requiring us to decipher the value of an unknown quantity. This article delves into a classic mathematical problem: if four times a number plus 3 is 11, what is the number? To effectively tackle this problem, we will embark on a journey through algebraic expressions, equations, and the art of isolating variables. Through a step-by-step approach, we will demystify the solution, empowering you to conquer similar mathematical challenges with confidence. Prepare to unlock the secrets hidden within this numerical enigma!
Understanding the Problem
Before diving into the solution, let's first grasp the essence of the problem. The question presents a mathematical statement that establishes a relationship between an unknown number and a series of operations. Our mission is to unravel this relationship and pinpoint the value of the elusive number. To do this effectively, we need to translate the verbal statement into a mathematical equation. This involves representing the unknown number with a variable, typically 'x', and expressing the operations of multiplication, addition, and equality using mathematical symbols. Once we have the equation, we can employ algebraic techniques to isolate the variable and determine its value.
Four times a number implies multiplying the number by 4, which can be written as 4x. Plus 3 indicates adding 3 to the result, leading to the expression 4x + 3. The statement is 11 signifies that the expression 4x + 3 is equal to 11, giving us the equation 4x + 3 = 11. Now that we have successfully translated the problem into a mathematical equation, we are well-equipped to embark on the journey of solving for x.
Setting up the Equation
The foundation of solving any algebraic problem lies in setting up the equation correctly. In our case, we are given the statement: "If four times a number plus 3 is 11." Let's break this down step-by-step to construct the equation.
First, we introduce a variable, typically 'x', to represent the unknown number. This variable will serve as a placeholder for the value we are trying to find. Next, we translate the phrase "four times a number" into mathematical notation. This simply means multiplying the variable 'x' by 4, resulting in the term 4x. The phrase "plus 3" indicates that we need to add 3 to the previous term, giving us 4x + 3. Finally, the statement "is 11" implies that the entire expression 4x + 3 is equal to 11. This leads us to the equation 4x + 3 = 11. This equation encapsulates the core of our problem, and by solving for 'x', we will unveil the hidden number. This crucial step of setting up the equation correctly is paramount to arriving at the accurate solution. It's like laying the foundation for a building; a strong foundation ensures a sturdy structure. Therefore, take your time to understand the problem statement and translate it accurately into a mathematical equation. With the equation in place, we are now ready to move on to the next phase: solving for the unknown variable.
Solving for the Unknown
With the equation 4x + 3 = 11 firmly established, we now embark on the exciting journey of solving for the unknown variable, 'x'. The goal is to isolate 'x' on one side of the equation, thereby revealing its value. To achieve this, we employ the fundamental principles of algebraic manipulation.
The first step involves neutralizing the constant term on the same side as 'x'. In our equation, this means getting rid of the '+ 3'. To do this, we perform the inverse operation, which is subtracting 3 from both sides of the equation. This ensures that the equation remains balanced, adhering to the golden rule of algebra: what you do to one side, you must do to the other.
Subtracting 3 from both sides gives us: 4x + 3 - 3 = 11 - 3. Simplifying this, we get 4x = 8. Now, 'x' is one step closer to being isolated. The next step is to eliminate the coefficient multiplying 'x'. In our case, 'x' is multiplied by 4. To counteract this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 4, maintaining the balance and preserving the equality.
Dividing both sides by 4, we have: 4x / 4 = 8 / 4. Simplifying this, we arrive at the solution: x = 2. Congratulations! We have successfully solved for 'x'. This methodical approach of applying inverse operations to isolate the variable is the cornerstone of solving algebraic equations. Each step is carefully executed to maintain the balance of the equation and gradually unveil the value of the unknown. With 'x' now revealed to be 2, we have cracked the code of our mathematical puzzle.
Verifying the Solution
In the realm of mathematics, verification is the cornerstone of ensuring accuracy. Before we declare victory and move on, it's crucial to validate our solution and confirm that it indeed satisfies the original problem statement. This process involves plugging the value we obtained for 'x' back into the original equation and verifying if both sides of the equation balance out. This step is not just a formality; it's a safeguard against potential errors and a testament to the rigor of mathematical thinking.
Our original equation is 4x + 3 = 11, and we found x = 2. Let's substitute this value back into the equation: 4 * 2 + 3 = 11. Now, we simplify the left-hand side of the equation: 8 + 3 = 11. Further simplification yields 11 = 11. Voila! The left-hand side of the equation equals the right-hand side, confirming that our solution x = 2 is indeed correct. This verification step provides us with the confidence that we have not only solved the equation but also understood the underlying mathematical relationship. It reinforces the importance of accuracy and attention to detail in problem-solving.
Conclusion
In this mathematical journey, we embarked on a quest to decipher the value of an unknown number within the equation 4x + 3 = 11. Through a systematic approach, we successfully navigated the realms of algebraic expressions, equations, and the art of isolating variables. We began by translating the verbal problem statement into a mathematical equation, laying the foundation for our solution. Then, we employed inverse operations to isolate the variable 'x', revealing its value to be 2. Finally, we rigorously verified our solution, ensuring its accuracy and solidifying our understanding of the problem.
This problem serves as a microcosm of the broader world of mathematics, where logical reasoning and methodical problem-solving reign supreme. The ability to translate real-world scenarios into mathematical equations and solve for unknowns is a valuable skill that extends far beyond the classroom. It empowers us to analyze data, make informed decisions, and tackle challenges in various aspects of life. As we conclude this exploration, let us carry forward the principles of accuracy, verification, and the joy of unraveling mathematical mysteries.
So, the answer to the question "If four times a number plus 3 is 11, what is the number?" is indeed 2.