Problem 54 - Algebra 1: Concepts and Skills

Question

In Exercises 53 and \(54,\) find and correct the error. $$ \begin{aligned} -2(7-8) &=-2(7)-2(8) \\ &=-14-6 \\ &=-30 \end{aligned} $$

Step-by-Step Solution

Answer

The correct result of the given equation is 2.

Step 1: Identify the error

The mistake is made in the first step when incorrectly distributing -2 over (7-8). The correct procedure should produce -2(7)+2(8), not -2(7)-2(8).

Step 2: Apply the correct Distributive law

The arithmetic rule that applies to this type of equation is the distributive property of multiplication over subtraction, which results in multiplying -2 by 7 and adding to it the result of multiplying -2 by -8. The equation should look like: -2*7 + -2*(-8).

Step 3: Calculate the correct results

By adopting the correct operations, it will be -14 + 16 which is equal to 2. Therefore, the correct result of the given equation is 2, not -30.

Key Concepts

Error Correction

Sometimes, errors can sneak into algebraic problems, especially when working with the distributive property. In this exercise, the error occurred during the initial step of distributing the multiplication factor across a subtraction in parentheses. As students, it's vital to recognize where errors might happen, so you can better identify and address them in your own work.

Keep an eye out for:

• Misapplying operations, such as distribution, where a sign might be mistakenly used.
• Confusing subtraction with addition during distribution, like treating \( -2 (7 - 8) \) as \( -2 \times 7 - 2 \times 8 \) instead of the correct operation which requires more attention to signs.
• The importance of correctly following order of operations, ensuring each step logically follows from the last.

Practicing these checks helps avoid mistakes and builds a stronger understanding of mathematical processes.

Algebraic Operations

Algebraic operations include basic operations such as addition, subtraction, multiplication, and division applied within an algebra context. They enable us to manipulate and solve equations efficiently.

Understanding the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is essential.

• Parentheses first ensures that operations within grouping symbols are done before anything else.
• Exponentiation next, though not needed in this problem, is frequently part of more complex equations.
• Multiplication or division follows, applied from left to right as they appear.
• Finally, addition or subtraction from left to right finalizes the solution.

In the context of the distributive property exercise, the correct application of multiplication across subtraction must adhere to these rules, ensuring \( -2 \times (7 - 8) \) becomes \( -2 \times 7 + -2 \times 8\). Adhering to these principles maintains consistency in finding accurate solutions.

Multiplication

In this particular problem, multiplication is central to using the distributive property effectively. It's essential to understand how multiplication works in algebraic contexts, especially when it interacts with other operations like addition or subtraction.

The distributive property itself is a specific rule about how multiplication distributes over addition or subtraction: \( a(b + c) = ab + ac \). But, it can also include negative numbers, as seen here.

• The rule applies not just to addition but also to subtraction, which is just adding a negative, shifting focus to \( a(b - c) = ab - ac \).
• Pay careful attention to the signs in your operation. Multiplying a negative number changes the signs post-distribution, often leading to errors if done incorrectly, like changing \( -2 \times -8 \) to \(+16\).
• Working through each multiplication separately simplifies the equation. Calculate, then balance these calculations, showing stepwise improvements.

Mastering multiplication with the distributive property is crucial for advancing in algebra and reliably solving problems.