Problem 112 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\begin{aligned} a(x-7)+b(7-x) &=a(x-7)+b(-1)(x-7) \\ &=a(x-7)-b(x-7) \\ &=(x-7)(a-b) \end{aligned}$$

Step-by-Step Solution

Answer

The original equation contains one false statement which is \(a(x-7)+b(7-x) = a(x-7)+b(-1)(x-7)\). The correct statement should be \(a(x-7) - b(x-7) = (x - 7)(a - b)\).

Step 1: Verify the first claim

Let's compare \(a(x-7)+b(7-x)\) to \(a(x-7)+b(-1)(x-7)\). We can observe that these expressions are not equivalent because when \(b(7-x)\) is replaced by \(b(-1)(x-7)\), it is necessary to be wary of the sign. In this case, it should be \(b(7-x) = -b(x-7)\) and not \(b(-1)(x-7)\). Therefore, this statement is false.

Step 2: Correct the false statement

The correct statement should be \(a(x-7) - b(x-7)\). Let's put it into the given equation replacing \(a(x-7)+b(-1)(x-7)\), we get \(a(x-7) - b(x-7)\). Now this statement is true.

Step 3: Verify the second claim

We now need to verify whether \(a(x-7) - b(x-7) = (x - 7)(a - b)\). By factoring out \((x-7)\) from the expression \(a(x - 7) - b(x - 7)\), we get \((x-7)(a-b)\). Therefore, this statement is true.

Key Concepts

Distributive Property

The distributive property is a cornerstone of algebra which allows us to multiply a single term by each term within a set of parentheses. Understanding this property is crucial for simplifying and factoring algebraic expressions. In our exercise, the distributive property comes into play when dealing with the expression \(a(x-7)+b(7-x)\).

To illustrate the distributive property, consider the example \(a(b + c)\). Instead of multiplying \(a\) by \(b + c\) as a grouped sum, we can distribute \(a\) across the terms inside the parentheses, giving us \(ab + ac\). Similarly, in the context of the given equation, we might expect to distribute the negative sign across the terms when \(b\) is multiplied by \(7-x\), resulting in \(b(7) - b(x)\) which simplifies to \(7b - bx\).

Factoring by Grouping

Factoring by grouping is a method used to factor complex expressions that do not easily lend themselves to other factoring methods. This technique involves rearranging and grouping terms in such a way that common factors can be extracted.

To use factoring by grouping, we generally look for terms that share a common factor and then group them accordingly. For example, if we have \(ax + ay + bx + by\), we can group the terms as \(ax + bx\) and \(ay + by\), then factor out the common terms \(x\) and \(y\) to get \(x(a + b) + y(a + b)\), and ultimately factor out the \(a + b\), resulting in \( (a + b)(x + y)\). In our original exercise, after correcting the false statement, the factoring by grouping method was correctly applied by factoring out the common \(x - 7\) term, leading to the expression \( (x-7)(a-b)\).

Verifying Algebraic Statements

Verifying algebraic statements is the process of determining the truthfulness of a given equation or expression. It is done by simplifying both sides of the equation and checking if the simplifications yield the same result.

In our exercise, the verification process was used at two key points. Initially, it was needed to scrutinize whether \(a(x-7)+b(7-x)\) could be represented as \(a(x-7)+b(-1)(x-7)\). Through verification, we discovered that the transformation was inaccurate due to the incorrect distribution of the negative sign. Following this, the second statement \(a(x-7) - b(x-7) = (x - 7)(a - b)\) was verified. After factoring, both sides of the equation matched, confirming that the factored form was indeed correct. Verification is essential as it ensures the steps taken during the simplification and factoring process preserve the equations’ integrity.

Correcting False Algebraic Statements

Correcting false algebraic statements is a critical skill in mathematics that involves identifying errors in reasoning or calculation and amending them to reach a true statement. Often, students may make common mistakes, such as mishandling signs or misapplying algebraic properties which lead to inaccurate expressions.

In the original problem, the incorrect application of the distributive property led to a false statement. Instead of properly distributing the negative sign through the expression \(b(7-x)\), it was mistakenly written as \(b(-1)(x-7)\). Recognizing and correcting this mistake was vital. The correction was to rewrite the term as \( -b(x-7)\), thus maintaining the integrity of the original expression. This shows how crucial attention to detail is when working with algebraic expressions and the importance of re-evaluating each step to ensure accuracy.