Problem 61 - Algebra A Combined Function

Question

$$ \text { In which of the following is } \frac{a+7}{7} \text { simplified correctly? (See the Concept Check in this section.) } $$ a. \(a+1\) b. \(a\) c. \(\frac{a}{7}+1\)

Step-by-Step Solution

Answer

Option c: \( \frac{a}{7} + 1 \) is the correct simplification.

Step 1: Understand the Expression

The expression provided is \( \frac{a+7}{7} \). We need to simplify it by considering the operations involved: division and how it affects both terms in the numerator.

Step 2: Distribute the Denominator

We distribute the denominator 7 to each term in the numerator \( a+7 \). This gives us two separate fractions: \( \frac{a}{7} + \frac{7}{7} \).

Step 3: Simplify the Fractions

Now simplify each fraction separately. \( \frac{a}{7} \) cannot be simplified further without additional information about 'a'. However, \( \frac{7}{7} \) simplifies to 1 because any number divided by itself is 1.

Step 4: Combine the Results

Combine the simplified fractions \( \frac{a}{7} + \frac{7}{7} \) to get the fully simplified expression \( \frac{a}{7} + 1 \).

Step 5: Match with the Options

Compare the simplified expression \( \frac{a}{7} + 1 \) to the options given: a) \( a+1 \), b) \( a \), c) \( \frac{a}{7}+1 \). The correct simplification matches option c.

Key Concepts

Understanding the Distributive Property

The distributive property is a fundamental principle in algebra that allows you to simplify expressions by distributing a factor across terms within parentheses. When you have an expression such as \( a(b + c) \), the distributive property lets you expand it to \( ab + ac \).
This rule is especially helpful in simplifying expressions and solving equations efficiently.

• When applying the distributive property, break down each term inside the parentheses by multiplying it with the term outside.
• This property holds true for subtraction as well, so \( a(b - c) = ab - ac \).

In the exercise, we see this principle in action by distributing the denominator across the terms in the numerator. By spreading the denominator \( 7 \) across the sum \( a + 7 \), we rewrite the fraction as \( \frac{a}{7} + \frac{7}{7} \). This step is key in breaking down and simplifying the expression further.

Working with Fractions

Fractions are numerical quantities represented as one part of a whole, split into equal segments. They're in the form \( \frac{numerator}{denominator} \), where the numerator is part of the whole, and the denominator is the total number of equal parts.
Fractions can often complicate algebraic expressions, but understanding their properties makes it easier to simplify or manipulate them.

• When every component of a fraction shares a common factor with the denominator, you can simplify the fraction by dividing each by that common factor.
• A fraction such as \( \frac{7}{7} \) simplifies to \( 1 \) since any number over itself equals \( 1 \).

In the given problem, the fractions created by distributing the denominator were \( \frac{a}{7} \), which remains as is without further simplification, and \( \frac{7}{7} \), which simplifies to \( 1 \). Handling each fraction separately streamlines the simplification process.

Simplifying Expressions Efficiently

Simplifying expressions in algebra makes them easier to work with and helps in solving equations faster. This involves condensing the expression into its simplest form by using mathematical operations and properties.
Here’s how you can simplify expressions effectively:

• First, distribute any common factors shared among the terms, like the denominator in fractions.
• Next, look for and perform any possible operations such as converting \( \frac{7}{7} \) to \( 1 \).
• Finally, combine like terms if there are any to form a neat, simplified expression.

In our example, distributing the denominator allowed us to split the original fraction into two simpler parts: \( \frac{a}{7} + 1 \). This resultant expression aligns with the correct choice provided, which is \( \frac{a}{7} + 1 \), illustrating efficient simplification. By applying these strategies, you can become proficient in handling complex algebraic expressions.