Problem 41 - Algebra A Combined Function

Question

Rewrite each rational expression as an equivalent rational expression with the given denominator. $$ \frac{9 a+2}{5 a+10}=\frac{\underline{\phantom{xx}}}{5 b(a+2)} $$

Step-by-Step Solution

Answer

The equivalent expression is \( \frac{9ab+2b}{5b(a+2)} \).

Step 1: Factor the original denominator

The original given expression is \( \frac{9a+2}{5a+10} \). We first factor the denominator, \( 5a + 10 \). Notice that \( 5a + 10 \) can be rewritten as \( 5(a + 2) \) since both terms in the denominator can be factored by the common factor of 5.

Step 2: Introduce the new denominator

The problem asks us to rewrite the expression with a new denominator: \( 5b(a+2) \). Notice that this is the original denominator \( 5(a+2) \), multiplied by \( b \). This means our fraction must be changed so that \( b \) is included in the denominator.

Step 3: Adjust the numerator accordingly

To adjust the numerator to match the new denominator, multiply the numerator by \( b \) as well. Thus, the numerator becomes \( b(9a+2) \). So, our expression now looks like \( \frac{b(9a+2)}{5b(a+2)} \).

Step 4: Expand and verify

To verify the solution, expand \( b(9a+2) \) to see if it is consistent with the equivalent transformed expression. Multiply \( b (9a) + b(2) \) to get \( 9ab + 2b \). Therefore, our new equivalent rational expression is \( \frac{9ab+2b}{5b(a+2)} \).

Key Concepts

Factoring in Rational Expressions

When dealing with rational expressions, factoring is a crucial first step. It helps simplify the expression or transform it into an equivalent form. Consider the expression \( \frac{9a+2}{5a+10} \). Before doing anything else, let's focus on the denominator, \( 5a+10 \). Factoring involves breaking down an expression into a product of simpler terms. Here, observe that both terms in the denominator share a common factor of 5. So, we can factor it out:

• Start with the entire expression \( 5a + 10 \).
• Notice that 5 is common, so factor it out: \( 5(a+2) \).

This step makes it easier to work with the rational expression because it's now in a simpler form. Identifying common factors is key and often involves basic arithmetic and pattern recognition. Always check to see if factoring further aids in transforming the expression.

Understanding Denominators

Denominators are a vital aspect of rational expressions since they define the division present in the expression. In the problem at hand, we are asked to convert the rational expression from a denominator of \( 5(a+2) \) to \( 5b(a+2) \). But why is this necessary? The task requires us to adapt our original expression to have this new denominator, implying an equivalent form exists with it. Notice the difference made by the new denominator:

• The original denominator: \( 5(a+2) \).
• The new denominator: \( 5b(a+2) \).

This highlights the importance of expanding or modifying the denominator while maintaining a balance in the fraction. To achieve equivalent transformations in expressions, any changes made to the denominator must also reflect in the numerator.

Numerator Adjustment for Equivalence

When the denominator changes, we need to adjust the numerator to maintain the equivalence of the rational expression. This adjustment ensures the value of the expression remains unchanged, despite the different forms. For the change from a denominator of \( 5(a+2) \) to \( 5b(a+2) \), the numerator must be modified to keep the expression equivalent. Here's how we perform the adjustment:

• Identify the required change; in this case, the insertion of \( b \) into the denominator requires a similar multiplication in the numerator.
• Our original numerator is \( 9a+2 \).
• Multiply it by \( b \) to align with the new denominator change: \( b(9a+2) \).
• Expand if necessary, \( b(9a+2) = 9ab + 2b \), to verify correctness.

This crucial step keeps the expressions 'equivalent,' meaning they represent the same mathematical situation, even if they are visually different.