Easy Math: Multiply -8(4y-7)

Welcome, math enthusiasts and learners! Today, we're diving into a fundamental concept in algebra: multiplying a binomial by a constant. Specifically, we'll be tackling the expression -8(4y - 7). This might look a little intimidating at first glance, but trust me, it's quite straightforward once you grasp the distributive property. We'll break down the process step-by-step, ensuring you understand why we do each step, not just how. Our goal is to make this concept clear and accessible, transforming any potential confusion into confidence. So, grab a pen and paper, or just follow along with your eyes, and let's unravel the mystery behind multiplying expressions like -8(4y - 7).

Understanding the Distributive Property

The heart of solving -8(4y - 7) lies in a crucial algebraic principle: the distributive property. In simple terms, the distributive property tells us that when you multiply a number by a sum or difference inside parentheses, you must multiply that number by each term within the parentheses. For our problem, -8 is the number outside the parentheses, and (4y - 7) is the expression inside. So, we need to distribute the -8 to both the 4y term and the -7 term. It's like sharing a treat with everyone in a room; the -8 has to interact with both 4y and -7. This property is fundamental in simplifying algebraic expressions and is a building block for more complex mathematical operations. Without it, expanding expressions like -8(4y - 7) would be impossible. Remember, the sign in front of the number being distributed is just as important as the number itself. In this case, the negative sign of the -8 will play a significant role in the final answer, affecting the signs of both terms inside the parentheses. Let's visualize this: imagine you have -8 groups, and each group contains 4y items and -7 items. To find the total number of items, you'd multiply -8 by 4y and then multiply -8 by -7, and finally, add those results together. This visual analogy helps solidify the concept of distribution and its application to expressions like -8(4y - 7).

Step-by-Step Solution for -8(4y - 7)

Now, let's apply the distributive property to our specific problem: -8(4y - 7). The first step is to multiply -8 by the first term inside the parentheses, which is 4y. Remember your rules for multiplying integers: a negative number multiplied by a positive number results in a negative number. So, -8 * 4y = -32y. This is our first part of the expanded expression. The second step is to multiply -8 by the second term inside the parentheses, which is -7. Here's where paying attention to signs is critical. A negative number multiplied by another negative number results in a positive number. Therefore, -8 * -7 = +56. Now, we combine the results from both multiplications. We have -32y from the first step and +56 from the second step. Putting them together, we get -32y + 56. This is the fully expanded and simplified form of the original expression -8(4y - 7). It's crucial to double-check each multiplication, especially the signs, to avoid common errors. Each term within the parentheses must be acted upon by the multiplier outside. If there were more terms inside, we would continue this process until every term had been multiplied by -8. The order of operations (PEMDAS/BODMAS) isn't directly applicable here in terms of resolving the parentheses first, as the expression inside is a variable term and a constant, which cannot be combined. Thus, the distributive property becomes the primary tool for simplification. This systematic approach ensures accuracy when dealing with expressions like -8(4y - 7).

Analyzing the Options

We've successfully distributed -8 across (4y - 7) and arrived at the answer -32y + 56. Now, let's look at the multiple-choice options provided to see which one matches our result. The options are:

a. -32y + 56 b. 32y - 56 c. -32y + 63 d. -32y - 56

Comparing our derived answer, -32y + 56, with the given options, we can see a direct match with option a. Option b has the correct coefficients but the wrong signs, likely resulting from incorrectly multiplying the negative numbers. Option c has an incorrect constant term, possibly due to a calculation error or misinterpreting the 7. Option d has the correct variable term but the wrong sign for the constant term, again stemming from sign errors during multiplication. It's always a good practice to review your work when presented with multiple choices, ensuring that your final answer aligns precisely with one of the options and that the other options represent plausible, but incorrect, outcomes from common mistakes. This verification step for -8(4y - 7) confirms our calculation and reinforces our understanding of the distributive property and integer multiplication rules.

Common Pitfalls and How to Avoid Them

When working with problems like -8(4y - 7), several common pitfalls can trip students up. The most frequent one is sign errors. As we saw, multiplying a negative by a positive yields a negative, while multiplying a negative by a negative yields a positive. Forgetting these basic rules when distributing the -8 is a primary cause of incorrect answers. For instance, incorrectly calculating -8 * -7 as -56 instead of +56 would lead to the wrong final expression. Another pitfall is not distributing to all terms. Some might multiply -8 by 4y but forget to multiply it by -7, or vice versa. Remember, the multiplier outside the parentheses must be applied to every single term inside. A third issue can be a simple arithmetic mistake in the multiplication itself, like calculating 8 * 4 as something other than 32. To avoid these pitfalls when solving -8(4y - 7):

1. Slow Down and Focus on Signs: Make a conscious effort to track the signs during each multiplication. You might even want to write down the sign rule (negative * positive = negative, negative * negative = positive) next to your problem as a reminder.
2. Underline or Circle Each Term: Before you start distributing, underline or circle each term inside the parentheses. Then, draw an arrow from the outside multiplier (-8) to each underlined term, visually representing the distribution. This helps ensure you don't miss any.
3. Double-Check Your Arithmetic: After you've performed the multiplications, quickly review the basic math facts involved (e.g., 8 * 4, 8 * 7) to catch any simple calculation errors.
4. Write Out the Intermediate Steps: Don't try to do too much in your head. Writing down each multiplication step (-8 * 4y = ... and -8 * -7 = ...) before combining them can prevent errors and makes it easier to find mistakes if you get the wrong answer.

By being mindful of these common errors and employing these strategies, you can confidently solve expressions like -8(4y - 7) and improve your overall algebraic skills.

Conclusion

We've successfully navigated the process of multiplying -8(4y - 7) using the distributive property. We learned that multiplying the outside constant -8 by each term inside the parentheses, 4y and -7, separately is the key. This resulted in -8 * 4y = -32y and -8 * -7 = +56. Combining these gave us our final answer: -32y + 56. This matches option a from the provided choices. Understanding and correctly applying the distributive property, along with careful attention to integer signs during multiplication, are essential skills for algebraic success. These principles extend far beyond this single problem, forming the basis for simplifying more complex expressions and equations. Keep practicing these fundamental concepts, and you'll find your confidence and accuracy grow. For further exploration into algebraic properties and solving equations, I recommend visiting Khan Academy's Algebra Section, a fantastic resource for in-depth explanations and practice exercises.