Solving The Math Problem The Difference Of A Number And 6
At the heart of mathematics lies the art of problem-solving, and word problems often present a unique challenge. They require us to translate real-world scenarios into mathematical expressions and equations, and then solve those equations to arrive at a solution. This particular problem, "The difference of a number and 6 is the same as 5 times the sum of the number and 2," is a classic example of such a puzzle. To effectively unravel this mystery, we need to dissect the problem statement, identify the key components, and then construct a mathematical equation that accurately represents the given information.
Breaking Down the Problem:
Let's begin by carefully examining the wording of the problem. The phrase "the difference of a number and 6" suggests a subtraction operation. We can represent the unknown number with a variable, say x. Thus, "the difference of a number and 6" can be written as x - 6. The phrase "is the same as" indicates equality, meaning we'll be setting up an equation. The next part, "5 times the sum of the number and 2," involves both addition and multiplication. The "sum of the number and 2" can be expressed as x + 2, and "5 times" this sum means we multiply the entire expression by 5, resulting in 5(x + 2). Now, we have all the pieces to construct our equation. We can connect these expressions using the "is the same as" phrase, which translates to an equals sign (=). So, the complete equation becomes:
x - 6 = 5(x + 2)
This equation is the mathematical representation of the problem statement. It encapsulates the relationships between the unknown number and the given constants. To find the value of the unknown number (x), we need to solve this equation. Solving equations is a fundamental skill in algebra, and it involves manipulating the equation using valid operations to isolate the variable on one side. This process often involves applying the order of operations, which dictates the sequence in which mathematical operations should be performed.
Navigating the Order of Operations:
The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial for simplifying expressions and solving equations correctly. In our equation, x - 6 = 5(x + 2), we first need to address the parentheses. This means we'll distribute the 5 across the terms inside the parentheses. Distributing the 5 means multiplying it by both x and 2. So, 5(x + 2) becomes 5x + 52, which simplifies to 5x* + 10. Now our equation looks like this:
x - 6 = 5x + 10
Next, we need to isolate the variable x. This means we want to get all the terms containing x on one side of the equation and all the constant terms (the numbers) on the other side. To do this, we can subtract x from both sides of the equation. This will eliminate the x term on the left side. Subtracting x from both sides gives us:
x - 6 - x = 5x + 10 - x
Simplifying this, we get:
-6 = 4x + 10
Now, we need to isolate the term with x. We can do this by subtracting 10 from both sides of the equation. Subtracting 10 from both sides gives us:
-6 - 10 = 4x + 10 - 10
Simplifying this, we get:
-16 = 4x
Finally, to isolate x, we need to divide both sides of the equation by 4. Dividing both sides by 4 gives us:
-16 / 4 = 4x / 4
Simplifying this, we get:
-4 = x
Therefore, the solution to the equation, and hence the answer to our word problem, is x = -4.
The Art of Verification:
After solving an equation, it's always a good practice to verify the solution. This involves substituting the value we found for the variable back into the original equation to see if it makes the equation true. In our case, we found x = -4. Let's substitute this back into the original equation:
x - 6 = 5(x + 2)
Substituting x = -4, we get:
-4 - 6 = 5(-4 + 2)
Simplifying the left side, we get:
-10 = 5(-4 + 2)
Simplifying the expression inside the parentheses on the right side, we get:
-10 = 5(-2)
Multiplying on the right side, we get:
-10 = -10
Since both sides of the equation are equal, our solution x = -4 is correct. This verification step gives us confidence in our answer and helps prevent errors.
The core of this mathematical challenge lies in the equation x - 6 = 5(x + 2). To unlock the value of x, we embark on a journey of algebraic manipulation, meticulously applying the principles of equation solving. The first step involves simplifying the equation by eliminating the parentheses. This is achieved by distributing the 5 across the terms inside the parentheses, a process that transforms 5(x + 2) into 5x + 10. Our equation now reads:
x - 6 = 5x + 10
Isolating the Unknown:
The next phase of our quest involves isolating the variable x. This means maneuvering the equation to have all terms containing x on one side and all constant terms on the other. A common strategy is to subtract x from both sides of the equation. This action elegantly eliminates x from the left side, paving the way for further simplification. Subtracting x from both sides transforms the equation into:
-6 = 4x + 10
The path to isolating x continues by addressing the constant term on the right side. We subtract 10 from both sides, effectively nullifying the +10 and bringing us closer to our goal. The equation now stands as:
-16 = 4x
The Final Unveiling:
The final act in our algebraic drama involves unveiling the value of x. To achieve this, we divide both sides of the equation by 4, the coefficient of x. This division isolates x, revealing its true value. The equation transforms into:
x = -4
Thus, we have successfully cracked the code, discovering that the unknown number x is -4. This value is not just a numerical answer; it's the solution to a mathematical puzzle, the culmination of our algebraic efforts.
The Significance of Each Step:
Each step in the equation-solving process is a deliberate maneuver, a calculated action designed to bring us closer to the solution. Distributing the 5 eliminates parentheses, clearing the way for further simplification. Subtracting x and 10 from both sides strategically isolates the variable, concentrating the unknown on one side of the equation. Dividing by 4 finally unveils the value of x, the culmination of our algebraic journey.
The order in which these steps are performed is not arbitrary. It follows the logical flow of algebraic principles, ensuring that each operation brings us closer to the solution without introducing errors. Mastering this sequence of operations is key to solving a wide range of algebraic equations, making it a fundamental skill in mathematics.
Through our meticulous problem-solving journey, we've arrived at the solution: the number is -4. Among the given options, this corresponds to option C. This is not just a numerical answer; it's the culmination of our efforts in translating a word problem into a mathematical equation and then solving that equation. The process we followed highlights the power of algebra as a tool for unraveling mathematical puzzles.
The Importance of Understanding the Process:
While arriving at the correct answer is satisfying, it's equally important to understand the process that led us there. Each step in the equation-solving process – distributing, isolating the variable, and performing arithmetic operations – is grounded in fundamental algebraic principles. These principles are not just abstract rules; they are the building blocks of mathematical reasoning. By understanding these principles, we can confidently tackle a wide range of mathematical problems.
Beyond the Single Problem:
The problem-solving skills we've honed in this exercise extend far beyond this particular question. They are applicable to various mathematical contexts and even to real-world scenarios that require logical thinking and problem-solving abilities. The ability to translate word problems into mathematical expressions, to manipulate equations, and to interpret solutions is a valuable asset in many fields, from science and engineering to finance and economics.
A Stepping Stone to Further Mathematical Exploration:
This problem serves as a stepping stone to further mathematical exploration. The concepts and techniques we've employed here form the foundation for more advanced algebraic topics. As we delve deeper into mathematics, we'll encounter more complex equations and problems, but the core principles of problem-solving will remain the same. The ability to dissect a problem, translate it into mathematical language, and then solve it is a skill that will serve us well throughout our mathematical journey.
The Beauty of Mathematical Problem-Solving:
Mathematical problem-solving is not just about finding the right answer; it's about the journey of discovery. It's about the satisfaction of unraveling a puzzle, of seeing the connections between different concepts, and of arriving at a solution through logical reasoning. This problem, "The difference of a number and 6 is the same as 5 times the sum of the number and 2," is a testament to the beauty and power of mathematical problem-solving.
Therefore, the correct answer is C. -4
