Master The Distributive Property: Equation Rewriting

Welcome to a deep dive into the fascinating world of algebra, where we'll unravel the mysteries of the distributive property! Today, we're tackling a specific challenge: how to correctly rewrite the equation $6(11-3)=4(7+5)$ using this fundamental mathematical concept. The distributive property, in essence, allows us to simplify expressions by multiplying a number outside a set of parentheses by each term inside those parentheses. It’s a powerful tool that helps us break down complex problems into more manageable steps. Think of it like distributing gifts to everyone in a room – each person gets something! We'll explore why this property is so crucial in mathematics and how applying it to our given equation leads us to the correct answer. Get ready to flex those algebraic muscles and gain a solid understanding of how to manipulate equations with confidence.

Understanding the Distributive Property in Detail

The distributive property is a cornerstone of arithmetic and algebra, and it's defined as $a(b+c) = ab + ac$. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. This principle extends to subtraction as well: $a(b-c) = ab - ac$. This property is incredibly useful because it allows us to simplify expressions, solve equations, and work with polynomials more efficiently. When we encounter an expression like $6(11-3)$, the number $6$ outside the parentheses needs to be distributed to both the $11$ and the $3$. This means we multiply $6$ by $11$ and then multiply $6$ by $3$. Crucially, we must also preserve the operation between these two resulting terms. Since the original expression inside the parentheses was a subtraction ($11-3$), the result of distributing the $6$ will also involve a subtraction: $6 imes 11 - 6 imes 3$. This is the essence of applying the distributive property. It's not just about multiplying; it's about maintaining the structure and relationships between the numbers and operations within the expression. Many students initially make the mistake of only distributing the number to the first term inside the parentheses, or they might incorrectly change the operation. However, a careful application ensures that the value of the expression remains unchanged. This property forms the basis for many algebraic manipulations, including factoring and expanding polynomials, making it indispensable for advanced mathematical study.

Applying the Distributive Property to the Equation

Now, let's get hands-on with our specific equation: $6(11-3)=4(7+5)$. Our goal is to rewrite both sides of this equation using the distributive property. Let's start with the left side: $6(11-3)$. Applying the distributive property, we multiply $6$ by $11$ and then subtract the product of $6$ and $3$. This gives us $6 imes 11 - 6 imes 3$. Performing these multiplications, we get $66 - 18$. Now, let's move to the right side of the equation: $4(7+5)$. Here, we distribute the $4$ to both the $7$ and the $5$. This means we multiply $4$ by $7$ and then add the product of $4$ and $5$. This results in $4 imes 7 + 4 imes 5$. Performing these multiplications, we get $28 + 20$. So, by correctly applying the distributive property to both sides of the original equation, we transform $6(11-3)=4(7+5)$ into $66 - 18 = 28 + 20$. This transformed equation is an equivalent representation of the original, achieved through the systematic application of the distributive property. It’s important to note that this step is purely about rewriting the expression, not necessarily solving for a variable (since there isn't one here). The focus is on demonstrating the understanding of how the distributive property alters the form of the expression while maintaining its mathematical value. This careful distribution ensures that the equality originally stated holds true after the transformation, reinforcing the fundamental rules of algebraic manipulation and the preservation of mathematical equivalence.

Evaluating the Options Provided

We've successfully rewritten the equation using the distributive property as $66 - 18 = 28 + 20$. Now, let's carefully examine the multiple-choice options provided to see which one matches our derived equivalent equation. We need to find the option that correctly represents the distribution on both sides of the original equation $6(11-3)=4(7+5)$.

• Option A: $18-66=20+28$: This option seems to have the terms on the left side reversed ($18-66$ instead of $66-18$) and the terms on the right side also reversed ($20+28$ instead of $28+20$). While the numbers themselves might be involved, the order of subtraction and addition is critical, and reversing them alters the value of the expression. Therefore, this option is incorrect.

• Option B: $66-18=28+20$: Let's compare this to our derived equation. On the left side, we have $66-18$, which matches our result from distributing $6$ to $11-3$. On the right side, we have $28+20$, which matches our result from distributing $4$ to $7+5$. This option perfectly aligns with our application of the distributive property. This is the correct answer.

• Option C: $66-33=28+5$: This option shows $66-33$ on the left. The $66$ comes from $6 imes 11$, but the $-33$ does not correctly result from distributing the $6$ to $-3$ (it should be $6 imes 3 = 18$, not $33$). The right side, $28+5$, is also incorrect; while $28$ might come from $4 imes 7$, the $5$ does not relate to the distribution of $4$ to $7+5$ (it should be $4 imes 5 = 20$). This option is incorrect.

• Option D: $66-3=28+35$: Similar to Option C, the left side has $66$, which is correct from $6 imes 11$. However, the $-3$ does not result from distributing $6$ to $-3$. The right side, $28+35$, is also incorrect. While $28$ could be from $4 imes 7$, the $35$ does not come from distributing $4$ to $7+5$ (it should be $4 imes 5 = 20$). This option is incorrect.

By systematically applying the distributive property and then evaluating each option against our derived equation, we can confidently identify Option B as the only correct representation.

The Importance of Precision in Algebraic Manipulation

This exercise highlights just how critical precision is when working with algebraic expressions and equations. The distributive property is a powerful tool, but it must be applied meticulously. A slight misstep – like forgetting to distribute to all terms inside the parentheses, reversing the order of operations, or performing incorrect multiplication – can lead to a completely wrong answer. In our case, the difference between the correct answer and the incorrect options often boiled down to small but significant errors: incorrect multiplication ($6 imes 3$ vs. $33$), incorrect distribution (only distributing to one term), or misordering terms in subtraction and addition. Understanding the underlying principle – that $a(b-c) = ab - ac$ and $a(b+c) = ab + ac$ – is the first step. The second, equally vital step, is the careful execution of these rules. Paying close attention to signs and numbers ensures that the transformed equation remains mathematically equivalent to the original. This precision is not just about getting the right answer on a test; it's about building a strong foundation for more complex mathematical concepts. Whether you're simplifying a physics formula, analyzing financial data, or developing software, the ability to manipulate equations correctly and reliably is a fundamental skill. Practice is key to developing this precision. The more you work through problems, the more intuitive the application of these properties will become, and the less likely you are to make careless errors. Remember, mathematics is a language, and like any language, clarity and accuracy in its grammar (the rules and properties) are essential for effective communication and problem-solving. Embracing this precision will serve you well in all your mathematical endeavors.

Conclusion

We have successfully navigated the process of rewriting the equation $6(11-3)=4(7+5)$ using the distributive property. By distributing the factor outside the parentheses to each term within, we transformed the equation into $66 - 18 = 28 + 20$. This transformation correctly represents the application of the distributive property to both sides of the original equation. When compared against the given options, it is clear that Option B: $66-18=28+20$ is the accurate rewrite. Mastering the distributive property is a key skill in algebra, enabling you to simplify expressions and solve equations more effectively. Always remember to distribute to every term inside the parentheses and maintain the original operations. Consistent practice with these types of problems will solidify your understanding and build your confidence in algebraic manipulation. For further exploration into the fundamental properties of algebra, you can visit Khan Academy's Algebra Section, a fantastic resource for learning and practice.