Problem 22 - Elementary Algebra

Question

For the following problems, fill in the missing term. $$ \frac{16}{4 y-16} $$

Step-by-Step Solution

Answer

Answer: The missing term in the fraction is (y-4).

Step 1: Identify common factors between 16 and (4y - 16)

Before simplifying the fraction, we need to determine if there are any common factors between the numerator (16) and the denominator (4y - 16). In this case, both terms have 4 as a common factor.

Step 2: Factor out the common factor 4 for both numerator and denominator

Since we identified 4 as the common factor, we will factor it out from both terms: $$ \frac{4 \cdot 4}{4( y - 4)} $$

Step 3: Simplify the fraction by canceling out the common factor

Now that we have factored out the common factor, we can cancel it, which leaves the missing term: $$ \frac{4}{ y - 4} $$ So, the missing term is \((y-4)\).

Key Concepts

Common Factors

When you're simplifying algebraic fractions, the term 'common factors' is fundamental. A common factor is a number or term that divides two or more numbers or terms without leaving a remainder. Think of it as a shared ingredient in a recipe.

• Identifying common factors is the key first step to simplifying fractions.
• Find these factors in both the numerator and the denominator.

In our exercise example, the number 4 is a common factor of both 16 and the expression \(4y - 16\). Recognizing this is the first move towards simplifying the given algebraic fraction.

Factoring

Once you identify common factors, the next step is to apply 'factoring.' Factoring involves expressing numbers or expressions as the product of their factors. Conceptually, it's about breaking down a number or an expression into simpler pieces.To factor the common factor out:

• Write the numerator and the denominator as products of this common factor.
• For instance, in our problem, 16 can be written as \(4 \times 4\), and \(4y - 16\) can be factored as \(4(y - 4)\).

This step is crucial because it's like unraveling a fraction into its basic components, which sets the stage for the next kind of simplification.

Canceling Common Factors

After factoring, one can proceed to the 'canceling' of common factors. This step simplifies the fraction.Here's how:

• Once you see the common factor in both the numerator and the denominator, you can "cancel" it out.
• Canceling involves removing the common factor from the fraction because anything divided by itself equals 1.

In our specific example, the common factor of 4, once factored, can be canceled, transforming the expression from \(\frac{4 \cdot 4}{4(y - 4)}\) to \(\frac{4}{y - 4}\). The end result is a simplified fraction where the original complexity is reduced to a simpler form, in this case, with the missing term \(y-4\). This transformation is a powerful tool in algebra, making expressions easier to handle and interpret.