Solving Twice The Difference Of A Number And 6 Is 7 An Algebraic Approach

Introduction: Understanding the Problem

In the realm of mathematics, we often encounter problems that require us to translate verbal statements into algebraic equations. These equations then become the tools we use to solve for unknown quantities. The problem presented, "Twice the difference of a number and 6 is 7," is a classic example of such a translation challenge. Our task is to dissect this sentence, identify the key mathematical operations and relationships, and express them in the concise language of algebra. We're specifically instructed to use the variable $y$ to represent the unknown number, adding another layer of precision to our translation. Before we dive into the solution, it's crucial to understand the importance of translating word problems into equations. This skill is fundamental to various fields, from engineering and physics to economics and computer science. The ability to represent real-world scenarios mathematically allows us to analyze complex situations, make predictions, and ultimately, solve problems.

This article will break down the given statement step-by-step, transforming it into a solvable algebraic equation. We'll explore the meaning of each phrase, the mathematical operations it implies, and how these pieces fit together to form the complete equation. By understanding this process, you'll gain a valuable tool for tackling similar problems in the future. The beauty of algebra lies in its ability to abstract real-world situations into symbolic representations. This abstraction allows us to manipulate and solve equations using established rules and techniques, ultimately leading us to the solution. So, let's embark on this journey of translation and unravel the mystery of the unknown number.

We will also discuss the steps involved in solving the equation once it is formed. Solving equations is a core skill in mathematics, and understanding the underlying principles is crucial for success. We'll explore the concept of inverse operations, which is the key to isolating the variable and finding its value. By applying these principles systematically, we can confidently solve a wide range of algebraic equations. Furthermore, we'll touch upon the importance of checking our solution. Verification is a critical step in the problem-solving process, ensuring that our answer is not only mathematically correct but also makes sense in the context of the original problem. This habit of checking solutions builds confidence and prevents errors from creeping into our work.

Breaking Down the Statement: Unpacking the Math

Let's carefully dissect the statement: "Twice the difference of a number and 6 is 7." To effectively translate this into an algebraic equation, we need to identify the mathematical operations and relationships embedded within the words. The phrase "a number" immediately suggests an unknown quantity, which we are told to represent with the variable $y$. This is our starting point – the foundation upon which we will build our equation. Next, we encounter the phrase "the difference of a number and 6." The word "difference" signals subtraction. It tells us that we need to subtract 6 from the unknown number, $y$. This gives us the expression $y - 6$. It is important to maintain the correct order of operations in subtraction, as changing the order will change the result. In this case, we are subtracting 6 from the number $y$.

Now, the statement says "twice the difference." The word "twice" indicates multiplication by 2. We are not simply multiplying the number $y$ by 2, but rather multiplying the entire expression representing the difference between $y$ and 6. This is where parentheses become crucial. We enclose the expression $y - 6$ in parentheses, and then multiply the entire quantity by 2. This ensures that we are taking two times the entire difference, not just the first term. So far, we have translated the phrase "twice the difference of a number and 6" into the algebraic expression $2(y - 6)$. This expression represents the left-hand side of our equation. It's the result of performing a series of mathematical operations on the unknown number $y$.

Finally, the statement concludes with "is 7." The word "is" acts as the great equalizer, the bridge that connects the left-hand side of our equation to the right-hand side. In mathematical language, "is" translates to the equals sign (=). This is a fundamental concept in algebra – the equals sign signifies that the expressions on either side of it represent the same value. So, "is 7" becomes "= 7". This completes the construction of our equation. We have successfully translated the verbal statement into a concise algebraic expression. The equation we have formed, $2(y - 6) = 7$, is the mathematical representation of the original problem. It encapsulates all the information provided in the statement and sets the stage for us to solve for the unknown number $y$.

Forming the Equation: From Words to Symbols

Having dissected the statement, we can now assemble the pieces into a complete algebraic equation. As we've established, "twice the difference of a number and 6" translates to $2(y - 6)$, and "is 7" translates to $= 7$. Combining these two parts, we arrive at the equation:

$2(y - 6) = 7$

This equation is the heart of the problem. It's a concise, symbolic representation of the original verbal statement. It captures all the essential mathematical relationships and allows us to use algebraic techniques to find the value of the unknown number, $y$. The process of translating word problems into equations is a cornerstone of mathematical problem-solving. It requires careful attention to detail, an understanding of mathematical vocabulary, and the ability to identify the underlying operations and relationships. This skill is not just confined to algebra; it's applicable across various branches of mathematics and other quantitative disciplines.

Now that we have our equation, the next step is to solve it. Solving an equation involves isolating the variable on one side of the equation, effectively revealing its value. This is achieved by performing inverse operations – operations that undo each other. We'll explore the process of solving this particular equation in detail in the next section. But before we move on, let's reflect on the significance of the equation itself. It's more than just a string of symbols; it's a powerful tool for representing and solving problems. It allows us to abstract real-world situations, manipulate them mathematically, and arrive at solutions that might not be immediately obvious.

The equation $2(y - 6) = 7$ represents a specific relationship between the unknown number $y$ and the number 7. It states that if we subtract 6 from $y$, and then multiply the result by 2, we will obtain 7. This relationship is what we will exploit to find the value of $y$. By systematically undoing the operations performed on $y$, we can peel away the layers and reveal the solution. This process is akin to reverse engineering – we're working backward from the final result to uncover the initial input.

Solving the Equation: Unveiling the Unknown

With the equation $2(y - 6) = 7$ in hand, we're ready to embark on the crucial step of solving for the unknown variable, $y$. The goal of solving an equation is to isolate the variable on one side, effectively determining its value. We achieve this by strategically applying inverse operations, which are operations that undo each other. The order in which we apply these operations is critical, and it often mirrors the reverse order of operations (PEMDAS/BODMAS). In our equation, the variable $y$ is first subjected to subtraction by 6, and then the result is multiplied by 2. Therefore, to isolate $y$, we need to reverse these operations – first, we'll undo the multiplication by 2, and then we'll undo the subtraction by 6.

The first step is to eliminate the multiplication by 2. The inverse operation of multiplication is division. To maintain the equality of the equation, we must perform the same operation on both sides. So, we divide both sides of the equation by 2:

$2(y - 6) / 2 = 7 / 2$

This simplifies to:

$y - 6 = 3.5$

Now, we have a simpler equation. The variable $y$ is only being subtracted by 6. To isolate $y$, we need to undo this subtraction. The inverse operation of subtraction is addition. We add 6 to both sides of the equation:

$y - 6 + 6 = 3.5 + 6$

This simplifies to:

$y = 9.5$

We have now successfully isolated $y$. The solution to the equation is $y = 9.5$. This means that the unknown number, when substituted back into the original equation, will make the equation true. This is a crucial concept – the solution to an equation is the value of the variable that satisfies the equation. It's the value that makes the left-hand side equal to the right-hand side.

Checking the Solution: Ensuring Accuracy

While we've solved for $y$, it's imperative to verify our solution. Checking the solution is a crucial step in the problem-solving process, as it helps us identify potential errors and ensures the accuracy of our answer. To check our solution, we substitute the value we found for $y$ (which is 9.5) back into the original equation:

$2(y - 6) = 7$

Substitute $y = 9.5$:

$2(9.5 - 6) = 7$

Now, we simplify the expression on the left-hand side:

$2(3.5) = 7$

$7 = 7$

The left-hand side of the equation is indeed equal to the right-hand side. This confirms that our solution, $y = 9.5$, is correct. It satisfies the original equation, making the statement "Twice the difference of 9.5 and 6 is 7" a true statement. The act of checking our solution provides us with confidence in our answer. It reinforces the understanding that we have correctly applied the algebraic principles and that our solution is not just a mathematical result but also a meaningful answer in the context of the problem.

Checking the solution also helps us catch potential errors that might have occurred during the solving process. Mistakes can happen at any stage – from translating the verbal statement into an equation to performing the algebraic manipulations. By substituting the solution back into the original equation, we create an independent check that can reveal these errors. If the left-hand side does not equal the right-hand side, it signals that a mistake has been made, and we need to revisit our steps to identify and correct the error.

Conclusion: The Power of Algebraic Translation

In conclusion, we have successfully translated the verbal statement "Twice the difference of a number and 6 is 7" into the algebraic equation $2(y - 6) = 7$, and we have solved for the unknown number, finding that $y = 9.5$. We also emphasized the importance of checking our solution, which confirmed the accuracy of our answer. This exercise highlights the power and versatility of algebra in representing and solving real-world problems. The ability to translate verbal statements into algebraic equations is a fundamental skill in mathematics and beyond.

This process involves careful analysis of the statement, identification of key mathematical operations and relationships, and the use of appropriate symbols and notation. It's a skill that requires practice and attention to detail, but the rewards are significant. Once we have an equation, we can apply algebraic techniques to solve for unknown quantities, make predictions, and gain insights into the relationships between variables. The equation becomes a powerful tool for analysis and problem-solving.

Furthermore, we've seen the importance of inverse operations in solving equations. By systematically undoing the operations performed on the variable, we can isolate it and reveal its value. This process is a cornerstone of algebraic manipulation and is applicable to a wide range of equations. The concept of inverse operations is not just a mechanical technique; it's a reflection of the underlying structure of mathematics, where operations have corresponding inverses that allow us to reverse their effects.

Finally, we underscored the critical role of checking our solution. This step is not just a formality; it's an essential part of the problem-solving process. It provides us with confidence in our answer, helps us catch potential errors, and reinforces our understanding of the underlying mathematical principles. Checking the solution is a habit that should be cultivated in every mathematical endeavor, as it ensures accuracy and promotes a deeper understanding of the concepts involved.

In essence, this problem exemplifies the power of algebraic translation – the ability to transform words into symbols, relationships into equations, and unknowns into solutions. This is a skill that empowers us to tackle a wide range of problems in mathematics, science, engineering, and beyond. As we continue to explore the world of mathematics, the ability to translate, solve, and verify will remain invaluable tools in our problem-solving arsenal.