Operation Used To Simplify 8(g-5)=48 To G-5=6

In the realm of algebra, solving equations is a fundamental skill. It involves manipulating equations while maintaining equality to isolate the variable of interest. A common technique is to perform the same operation on both sides of the equation, ensuring the balance is preserved. This article delves into a specific equation simplification, dissecting the operation employed by a student to transform the equation $8(g-5)=48$ into $g-5=6$. Understanding this process is crucial for students venturing into the world of algebra and equation-solving.

Decoding the Equation Transformation: From $8(g-5)=48$ to $g-5=6$

Let's embark on a journey to decipher the algebraic maneuver executed by the student. The starting point is the equation $8(g-5)=48$. Our objective is to isolate the term $(g-5)$ on one side of the equation. To achieve this, we need to eliminate the coefficient 8 that is multiplying the parentheses. The golden rule of equation solving dictates that any operation performed on one side must be mirrored on the other side to maintain the equation's balance. Now, let's analyze the options provided:

• A. Subtracted 8 from both sides of the equation: If we subtract 8 from both sides of the equation $8(g-5)=48$, we would get $8(g-5) - 8 = 48 - 8$, which simplifies to $8g - 40 - 8 = 40$, and further to $8g - 48 = 40$. This result does not align with our target equation of $g-5=6$, so subtraction is not the operation we're looking for.
• B. Divided by 8 on both sides of the equation: This operation holds promise. If we divide both sides of the equation $8(g-5)=48$ by 8, we obtain $\frac{8(g-5)}{8} = \frac{48}{8}$. Simplifying this yields $g-5 = 6$, precisely the equation we were aiming for. Division appears to be the key to this transformation.
• C. Multiplied by 8 on both sides of the equation: Multiplying both sides of the original equation by 8 would result in $64(g-5) = 384$, which is a far cry from our target equation. Multiplication would only complicate the equation further.
• D. Added 8 to both sides of the equation: Adding 8 to both sides of the original equation gives us $8(g-5) + 8 = 48 + 8$, which simplifies to $8g - 40 + 8 = 56$, and further to $8g - 32 = 56$. Again, this deviates from our desired outcome.

Therefore, after careful consideration, the correct answer is undeniably B. Divided by 8 on both sides of the equation. This operation effectively isolates the term $(g-5)$, paving the way for further simplification and solution of the equation.

Why Division is the Key to Isolating Variables in Equations

In the grand scheme of algebra, division plays a pivotal role in isolating variables within equations. It's the inverse operation of multiplication, and when applied strategically, it can unravel complex expressions. Consider an equation where a variable is multiplied by a coefficient, as in our example with $8(g-5)=48$. To liberate the variable from the clutches of the coefficient, we employ division. By dividing both sides of the equation by the coefficient, we effectively neutralize its effect on the variable, bringing us closer to the solution.

The principle of maintaining equality is paramount in this process. Whatever we do to one side of the equation, we must replicate on the other side. This ensures that the equation remains balanced, preserving the relationship between the variables and constants. Division, when executed with precision, allows us to systematically peel away the layers surrounding the variable, ultimately revealing its value.

Moreover, division often simplifies equations, making them more manageable and easier to solve. By reducing coefficients and constants, we create a clearer path towards the solution. In our specific example, dividing both sides by 8 transformed the equation from a somewhat cumbersome form to a much more streamlined version, $g-5=6$. This simplification is a testament to the power of division in the realm of algebraic manipulation.

The Significance of Understanding Inverse Operations in Algebra

Algebra, at its core, is a symphony of operations, each with its inverse counterpart. Addition dances with subtraction, while multiplication waltzes with division. Understanding these inverse relationships is the key to unlocking the secrets of equation solving. In essence, inverse operations undo each other, allowing us to strategically manipulate equations and isolate variables.

Consider addition and subtraction. If we have an equation where a constant is added to a variable, we can employ subtraction to nullify the constant's effect. Conversely, if a constant is subtracted from a variable, addition comes to the rescue. This dance of addition and subtraction is fundamental to isolating variables and simplifying equations.

Similarly, multiplication and division form an inseparable pair. When a variable is multiplied by a coefficient, division steps in to break the bond. Conversely, if a variable is divided by a constant, multiplication restores its original form. This interplay between multiplication and division is the backbone of many algebraic manipulations.

Mastering inverse operations is not merely a matter of rote memorization; it's about developing a deep understanding of how these operations interact. It's about recognizing the patterns and applying the appropriate inverse operation to achieve a desired outcome. This mastery empowers students to tackle a wide range of algebraic problems with confidence and precision.

Common Pitfalls to Avoid When Simplifying Equations

Navigating the world of equation simplification can be tricky, and there are several common pitfalls that students often encounter. Awareness of these pitfalls is crucial to avoid errors and ensure accurate solutions. One frequent mistake is neglecting to perform the same operation on both sides of the equation. The golden rule of equation solving is that any action taken on one side must be mirrored on the other. Failure to adhere to this principle leads to an imbalance and an incorrect solution.

Another common error lies in misapplying the order of operations. Remember the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Performing operations in the wrong order can lead to a cascade of errors. For instance, if we have an expression like $2 + 3 \times 4$, we must perform the multiplication before the addition, yielding $2 + 12 = 14$, not $5 \times 4 = 20$.

Furthermore, students sometimes struggle with distributing operations correctly. When faced with an expression like $a(b + c)$, we must distribute the 'a' to both 'b' and 'c', resulting in $ab + ac$. Neglecting to distribute properly can lead to significant errors in the simplification process.

Lastly, a lack of attention to signs can also derail equation solving. A misplaced negative sign can completely alter the outcome. It's crucial to carefully track signs throughout the simplification process, ensuring that they are applied correctly.

By being mindful of these common pitfalls and practicing diligently, students can hone their equation-solving skills and navigate the world of algebra with greater confidence.

Conclusion: Mastering Algebraic Simplification for Mathematical Success

In conclusion, the student adeptly employed division to simplify the equation $8(g-5)=48$ to $g-5=6$. This operation, the inverse of multiplication, effectively isolated the term $(g-5)$, paving the way for further solution of the equation. Understanding inverse operations and the principle of maintaining equality is paramount in the realm of algebraic manipulation. By mastering these concepts and avoiding common pitfalls, students can unlock the power of algebra and achieve mathematical success.