Solving Math Problems The Difference Of A Number And 6

Let's embark on a journey to decipher this intriguing mathematical puzzle. The problem presents a scenario where the difference between a number and 6 is equal to 5 times the sum of the number and 2. Our mission is to identify the elusive number that satisfies this condition. To solve this, we need to translate the word problem into a mathematical equation and then solve for the unknown variable.

The cornerstone of solving such mathematical word problems lies in translating the verbal statements into algebraic expressions. This process transforms the narrative into a concise and manageable mathematical form, allowing us to apply algebraic techniques to find the solution. In this case, we will represent the unknown number with a variable, typically denoted as 'x'. Then, we carefully deconstruct the problem statement, piece by piece, converting each phrase into its corresponding mathematical equivalent.

The phrase "the difference of a number and 6" translates directly into the algebraic expression x - 6. This represents the subtraction of 6 from the unknown number. The term "is the same as" signifies equality, which we represent with the equals sign (=). The phrase "5 times the sum of the number and 2" is slightly more complex. First, we address "the sum of the number and 2," which is simply x + 2. Then, we multiply this entire sum by 5, giving us 5(x + 2). By piecing together these components, we create the complete algebraic equation that accurately represents the problem statement.

Now that we have successfully translated the problem into an algebraic equation, our next step is to solve for the unknown variable, x. This involves applying the principles of algebra to isolate x on one side of the equation. Our equation, x - 6 = 5(x + 2), requires us to first address the parentheses on the right side. We do this by applying the distributive property, which involves multiplying the 5 by each term inside the parentheses. This transforms the equation into x - 6 = 5x + 10. With the parentheses removed, we can now proceed to collect like terms, bringing all the x terms to one side and the constant terms to the other.

To achieve this, we subtract x from both sides of the equation, which gives us -6 = 4x + 10. Then, we subtract 10 from both sides to isolate the term with x, resulting in -16 = 4x. Finally, to solve for x, we divide both sides by 4, leading us to the solution x = -4. This algebraic manipulation demonstrates how we systematically isolate the unknown variable, step by step, until we arrive at its numerical value. The process emphasizes the importance of maintaining balance on both sides of the equation, ensuring that each operation performed preserves the equality.

Now, let's translate the problem into an equation. If the number is x, then the difference of the number and 6 is x - 6. Five times the sum of the number and 2 is 5(x + 2). So, the equation is:

x - 6 = 5(x + 2)

Next, we need to solve this equation for x. First, distribute the 5 on the right side:

x - 6 = 5x + 10

Now, subtract x from both sides:

-6 = 4x + 10

Subtract 10 from both sides:

-16 = 4x

Finally, divide by 4:

x = -4

To ensure the accuracy of our solution, we must verify that the value x = -4 satisfies the original equation. This process involves substituting -4 for x in the equation x - 6 = 5(x + 2) and confirming that both sides of the equation yield the same result. This verification step is crucial as it helps us catch any potential errors made during the algebraic manipulation process. By substituting the value and simplifying both sides, we can confidently confirm that our solution is correct and that the number -4 indeed fulfills the conditions outlined in the problem statement.

Substituting x = -4 into the equation gives:

-4 - 6 = 5(-4 + 2)

-10 = 5(-2)

-10 = -10

The left side equals the right side, so the solution is correct.

Therefore, the number is -4. So the answer is:

A. -4

This problem illustrates how to translate a word problem into an algebraic equation and solve it step by step. By carefully following the algebraic principles, we were able to find the unknown number that satisfies the given condition. This approach is fundamental in solving various mathematical problems, emphasizing the importance of understanding the relationship between verbal statements and their corresponding mathematical expressions.

To truly master the art of solving mathematical word problems like this one, it's crucial to grasp the underlying concepts and principles that govern the process. The journey from a narrative problem statement to a numerical solution involves several key steps, each requiring a specific skill set and understanding. These steps include translating verbal phrases into algebraic expressions, setting up equations, applying algebraic techniques to solve for unknowns, and verifying the solution to ensure its accuracy. By delving into these core concepts, we can develop a robust framework for tackling a wide range of mathematical challenges.

One of the foundational concepts is the ability to represent unknown quantities with variables. This allows us to transform the abstract relationships described in the problem into concrete mathematical expressions. For instance, in this problem, we represented the unknown number with the variable x. This simple yet powerful technique enables us to manipulate and solve for the unknown, bridging the gap between the narrative and the mathematical realm. The choice of variable is often arbitrary, but consistency in its usage is key to maintaining clarity throughout the solution process. The variable acts as a placeholder, allowing us to perform operations and track the unknown quantity as we move towards the solution.

Translating verbal phrases into algebraic expressions is another critical skill. This involves recognizing the mathematical operations implied by various words and phrases. For example, "the difference of" indicates subtraction, "the sum of" indicates addition, "times" indicates multiplication, and "is the same as" indicates equality. By carefully dissecting the problem statement and identifying these key phrases, we can construct the algebraic equation that accurately represents the problem. This process requires a keen eye for detail and a solid understanding of mathematical vocabulary. The ability to translate words into algebra is not only essential for solving word problems but also forms the basis for more advanced mathematical concepts.

Setting up the equation correctly is paramount to finding the right solution. The equation acts as the mathematical model of the problem, encapsulating the relationships between the known and unknown quantities. A well-constructed equation accurately reflects the information provided in the problem statement, allowing us to apply algebraic techniques to find the solution. In this case, the equation x - 6 = 5(x + 2) captures the essence of the problem, stating that the difference between the number and 6 is equal to five times the sum of the number and 2. The equation serves as the roadmap for our solution, guiding us through the algebraic steps towards the final answer.

Once the equation is set up, the next step is to solve for the unknown variable. This requires mastering various algebraic techniques, such as the distributive property, combining like terms, and isolating the variable. The distributive property, as applied in this problem, allows us to simplify expressions involving parentheses. Combining like terms involves grouping terms with the same variable or constant, simplifying the equation and making it easier to solve. Isolating the variable is the ultimate goal of the solving process, where we manipulate the equation to get the variable alone on one side, revealing its value.

Finally, verifying the solution is an essential step in the problem-solving process. This involves substituting the solution back into the original equation to ensure that it satisfies the given conditions. Verification helps us catch any errors made during the algebraic manipulations and gives us confidence in our answer. In this problem, substituting x = -4 into the equation confirmed that both sides were equal, validating our solution. Verification is not just a procedural step but a critical component of mathematical reasoning, fostering accuracy and precision.

In conclusion, solving mathematical word problems like this requires a multifaceted approach, encompassing translation, equation setup, algebraic manipulation, and verification. By understanding the core concepts and mastering the necessary techniques, we can unlock the power of mathematical problem-solving and apply it to a wide range of real-world scenarios. This problem serves as a valuable exercise in developing these skills, highlighting the importance of precision, logic, and attention to detail. The journey from problem statement to solution is not just about finding the answer but about developing a deeper understanding of mathematical principles and their applications. This understanding empowers us to tackle more complex challenges and appreciate the elegance and power of mathematics.