Problem 47 - Algebra A Combined Function

Question

Perform the indicated operations. $$ \frac{5 x}{7}+\frac{9 x}{7} $$

Step-by-Step Solution

Answer

The result is \( 2x \).

Step 1: Identify Common Denominator

Both fractions, \( \frac{5x}{7} \) and \( \frac{9x}{7} \), share the same denominator, 7. This means we can directly combine the numerators.

Step 2: Combine Numerators

Since the denominators are the same, add the numerators together: \( 5x + 9x = 14x \).

Step 3: Form the Single Fraction

Put the combined numerator over the common denominator: \( \frac{14x}{7} \).

Step 4: Simplify the Fraction

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is 7: \( \frac{14x}{7} = 2x \).

Key Concepts

Common Denominator

When working with rational expressions, identifying a common denominator is vital for adding or subtracting fractions. The denominator is the number beneath the fraction bar that tells you how many equal pieces the whole is divided into. Having a common denominator means that the fractions you are working with have the same number of pieces or divisions. In the given problem, both fractions \( \frac{5x}{7} \) and \( \frac{9x}{7} \) already share a common denominator of 7. This simplifies our work because we can focus solely on adding the numerators.

• If fractions have different denominators, you need to find a least common denominator (LCD) before proceeding with operations like addition or subtraction.
• An easy way to find a common denominator is to multiply the denominators together, although this is not always the least. Hence, using the least common denominator makes calculations simpler.

By ensuring a common denominator, you streamline the process of combining fractions, making it a straightforward task.

Combining Fractions

Once you establish a common denominator, the next step is to combine the fractions. This involves adding or subtracting the numerators, the top parts of the fractions, while the common denominator remains unchanged. In the exercise mentioned:

• The fractions \( \frac{5x}{7} \) and \( \frac{9x}{7} \) have the same denominator, so we add the numerators: \( 5x + 9x \).
• This results in a new numerator, which is \( 14x \).

The combined fraction that we get is \( \frac{14x}{7} \). At this point, you have successfully unified the initial fractions into a single rational expression. Always remember to only operate on the numerators; the denominator stays constant throughout this operation. By combining fractions efficiently, you lay the groundwork for moving on to the simplification step.

Simplifying Fractions

The last, but crucial, step when working with rational expressions is simplifying the fractions. Simplification involves making the fraction as simple as possible by reducing the numerator and denominator down by their greatest common divisor (GCD). For the example \( \frac{14x}{7} \):

• The GCD of 14 and 7 is 7.
• To simplify, divide both the numerator and the denominator by 7, resulting in \( 2x \).

Simplifying fractions makes equations easier to handle and understand, especially in complex mathematical contexts. Always look for the largest number that both the numerator and the denominator can be divided by evenly. If there are variables, factor them out too if possible. By ensuring your fractions are in their simplest form, you confirm the precision and clarity of your solution.