Problem 10
Question
Complete the solution. Are the student's answers and the book's answers equivalent? $$ \begin{array}{|c|c|l|} \hline \text { Student's answer } & \text { Book's answer } & \text { Equivalent? } \\ \hline \frac{m^{2}+2 m}{(m-1)(m-4)} & \frac{m^{2}+2 m}{(m-4)(m-1)} & \\ \hline \frac{-5 x^{2}-7}{4 x(x+3)} & -\frac{5 x^{2}-7}{4 x(x+3)} & \\ \hline \frac{-2 x}{x-y} & -\frac{2 x}{x-y} & \\ \hline \end{array} $$
Step-by-Step Solution
Verified Answer
Yes, all of the student's answers are equivalent to the book's answers.
1Step 1: Evaluate the First Answers
The student's answer is \( \frac{m^{2}+2m}{(m-1)(m-4)} \) and the book's answer is \( \frac{m^{2}+2m}{(m-4)(m-1)} \). Both expressions represent the same rational expression because multiplication is commutative, meaning the order of factors does not affect the product. Therefore, the answers are equivalent.
2Step 2: Evaluate the Second Answers
The student's answer is \( \frac{-5x^{2}-7}{4x(x+3)} \) and the book's answer is \( -\frac{5x^{2}-7}{4x(x+3)} \). Simplifying the student's answer, \( \frac{-5x^{2}-7}{4x(x+3)} = -\frac{5x^{2}+7}{4x(x+3)} \) because the negative can be factored out of the numerator. This shows the two forms are equivalent.
3Step 3: Evaluate the Third Answers
The student's answer is \( \frac{-2x}{x-y} \) and the book's answer is \( -\frac{2x}{x-y} \). The expressions are identical as both represent the same expression when the negative is positioned outside or inside the fraction. Thus, they are equivalent.
Key Concepts
Commutative PropertyAlgebraic EquivalenceFraction Simplification
Commutative Property
The commutative property is a fundamental concept in mathematics. It states that the order in which two numbers are multiplied does not change the product. This property is incredibly useful when working with algebraic expressions, particularly with rational expressions.
\[ a \cdot b = b \cdot a \]
In the context of rational expressions, this property allows us to rearrange the factors in both the numerators and denominators without altering the value of the expression. For example, in the given exercise, the student's answer and the book's answer for the first problem are \( \frac{m^{2}+2m}{(m-1)(m-4)} \) and \( \frac{m^{2}+2m}{(m-4)(m-1)} \) respectively. Although the expressions look different at first glance, they are equivalent because of the commutative property.
This property helps ensure that, as long as you are multiplying, you can reorder factors to suit your needs without worrying about changing the essence of the expression. This flexibility makes simplifying expressions and solving equations much easier.
\[ a \cdot b = b \cdot a \]
In the context of rational expressions, this property allows us to rearrange the factors in both the numerators and denominators without altering the value of the expression. For example, in the given exercise, the student's answer and the book's answer for the first problem are \( \frac{m^{2}+2m}{(m-1)(m-4)} \) and \( \frac{m^{2}+2m}{(m-4)(m-1)} \) respectively. Although the expressions look different at first glance, they are equivalent because of the commutative property.
This property helps ensure that, as long as you are multiplying, you can reorder factors to suit your needs without worrying about changing the essence of the expression. This flexibility makes simplifying expressions and solving equations much easier.
Algebraic Equivalence
Algebraic equivalence refers to the idea that two expressions are equal if they have the same value for all values of the variables involved. This concept is pivotal in solving equations and verifying solutions in algebra because it confirms that different-looking expressions can represent the same mathematical relationship.
In the exercises provided, two rational expressions may appear different at first. However, upon closer examination, they are algebraically equivalent. Take, for instance, \( \frac{-5x^{2}-7}{4x(x+3)} \) (student's answer) and \( -\frac{5x^{2}-7}{4x(x+3)} \) (book's answer). They might look different due to the placement of the negative sign, but when simplified properly, these two forms clearly demonstrate algebraic equivalence.
Understanding this concept requires careful analysis of the expression components to determine if they are truly the same in value despite visual differences. It encourages students to look beyond the surface and analyze the underlying structures of algebraic expressions.
In the exercises provided, two rational expressions may appear different at first. However, upon closer examination, they are algebraically equivalent. Take, for instance, \( \frac{-5x^{2}-7}{4x(x+3)} \) (student's answer) and \( -\frac{5x^{2}-7}{4x(x+3)} \) (book's answer). They might look different due to the placement of the negative sign, but when simplified properly, these two forms clearly demonstrate algebraic equivalence.
Understanding this concept requires careful analysis of the expression components to determine if they are truly the same in value despite visual differences. It encourages students to look beyond the surface and analyze the underlying structures of algebraic expressions.
Fraction Simplification
Fraction simplification involves reducing a fraction to its simplest form, where the numerator and the denominator share no common factors other than one. With rational expressions, simplification plays a critical role to both solve and interpret mathematical equations correctly.
Simplifying rational expressions may involve factoring polynomials in the numerators or denominators, canceling out common factors, or rearranging terms to further see their relationships. For example, when we notice \( \frac{-2x}{x-y} \) and \( -\frac{2x}{x-y} \), simplification and understanding fraction manipulation help show these are actually the same due to the negative being factored or distributed.
Simplifying rational expressions may involve factoring polynomials in the numerators or denominators, canceling out common factors, or rearranging terms to further see their relationships. For example, when we notice \( \frac{-2x}{x-y} \) and \( -\frac{2x}{x-y} \), simplification and understanding fraction manipulation help show these are actually the same due to the negative being factored or distributed.
- Identify any common factors in the numerator and denominator.
- Cancel these factors out to simplify the expression.
- Always check to ensure the expression is in its simplest form.
Other exercises in this chapter
Problem 10
Simplify each complex fraction. See Example \(1 .\) $$ \frac{\frac{3}{5}}{\frac{2}{7}} $$
View solution Problem 10
By what should both sides of the equation be multiplied to clear it of fractions? $$ \text { a. } \frac{x}{5}=\frac{3 x}{10}+\frac{7}{2 x} $$ $$ \text { b. } \f
View solution Problem 10
Fill in the blanks. To write \(\frac{x}{x-9}\) as an equivalent rational expression with a denominator of \(3 x(x-9),\) we need to multiply the denominator by _
View solution Problem 10
b. Fill in the blank: \(\frac{1 \text { day }}{24 \text { hours }}=\)
View solution