Problem 102 - Intermediate Algebra

Question

After solving a formula for \(m,\) a student compared her answer with that at the back of the textbook. Could this problem have two different-looking answers? Explain why or why not. $$ \begin{aligned} &\text { Student: } m=\frac{5}{9} a r+1\\\ &\text { Book: } m=\frac{5 a r+9}{9} \end{aligned} $$

Step-by-Step Solution

Answer

Yes, the expressions are equivalent though they look different due to algebraic manipulation.

Step 1: Understand the Student's Answer

The student has expressed their answer for \( m \) as \( m = \frac{5}{9} ar + 1 \). This means they have assumed that the expression has two separate terms: \( \frac{5}{9} ar \) and \( 1 \).

Step 2: Understand the Book's Answer

The book has expressed the answer for \( m \) as \( m = \frac{5ar + 9}{9} \). This writes \( m \) as a single fraction, where the numerator is \( 5ar + 9 \) and the denominator is \( 9 \).

Step 3: Combine the Student's Expression into a Single Fraction

Rewrite the student's answer by combining into a single fraction: \( m = \frac{5}{9} ar + 1 \). Rewriting \( 1 \) as \( \frac{9}{9} \), the expression becomes \( m = \frac{5}{9} ar + \frac{9}{9} \). Combining these gives \( m = \frac{5ar + 9}{9} \), which matches the book's answer.

Step 4: Analyze Differences in Presentation

Although the forms look different, the expressions are equivalent. Sometimes, different algebraic manipulations lead to different appearances of the answer, even though their values remain the same when simplified.

Step 5: Conclusion of Equivalence

Both the student's solution and the book's solution are equivalent because they represent the same numerical value when simplified, just expressed in different forms.

Key Concepts

Equivalent Expressions

Understanding equivalent expressions is key in algebra. Even though expressions can look different, they can represent the same value. This is often seen in situations where an expression can be simplified or rearranged.

Consider the student's expression:

• She writes it as two separate terms: \( m = \frac{5}{9} ar + 1 \)
• The book simplifies it to a single fraction: \( m = \frac{5ar + 9}{9} \)

Both forms are equivalent because, when simplified, they yield the same number.

This principle is crucial because it highlights that different approaches can give you equivalent results. As long as the underlying math is correct, multiple different-looking expressions can be equally valid representations of the same value.

Fraction Simplification

Fraction simplification is the process of modifying fractions to their simplest form, making them easier to handle and compare. When comparing fractions, simplifying them can reveal how different expressions can actually be the same.

In the exercise, the fraction \( m = \frac{5}{9} ar + 1 \) needs simplification to see the equivalence to the book's form. Here’s how you can simplify it:

• Recognize that \( 1 \) can be written as a fraction over the same denominator, \( \frac{9}{9} \)
• This transforms the student's answer to \( m = \frac{5}{9} ar + \frac{9}{9} \)
• Combine the terms into a single fraction: \( m = \frac{5ar + 9}{9} \)

Now, the expressions look identical, simplifying the comparison and confirming their equivalence. Simplifying fractions effectively helps in reducing complex algebraic expressions to their simplest, most workable forms.

Algebraic Manipulation

Algebraic manipulation refers to the process of using algebraic rules and operations to rearrange and simplify expressions. It’s an essential skill for solving equations and understanding equivalencies among expressions.

The student's and the book's expressions are manipulated through operations like:

• Rewriting constants as fractions (from \( 1 \) to \( \frac{9}{9} \))
• Combining two fractions into one common denominator expression

These steps show how expressions that look different at first can be shown to be identical, demonstrating the power of manipulation.

Mastery of algebraic manipulation allows you to tackle more complex problems by simplifying and rearranging expressions where needed. It's a skill that gives you flexibility and control over how you solve algebraic equations and interpret different forms of expressions. With practice, it becomes an intuitive part of solving any algebra problem.