Problem 33 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{x}{7}+\frac{5}{3}$$

Step-by-Step Solution

Answer

So, the simplified form of the expression is \(\frac{3x+35}{21}\)

Step 1: Identify the Denominators

In the given expression, the denominators are 7 and 3.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator is found by determining the least common multiple (LCM) of the denominators. The LCM of 7 and 3 is 21.

Step 3: Rewrite the Fractions using the LCD

To do this, multiply the numerator and denominator of each fraction by the number needed to change each denominator to the LCD. So the expression becomes \(\frac{x*3}{7*3}+ \frac{5*7}{3*7}\) which simplifies to \(\frac{3x}{21}+\frac{35}{21}\).

Step 4: Combine the Fractions

Since the fractions now have the same denominator, they can be combined. The result is \(\frac{3x+35}{21}\).

Key Concepts

Least Common Denominator

When dealing with fractions, especially in algebra, it's crucial to work with a common playing field for the denominators. This common field is known as the Least Common Denominator (LCD). To find the LCD, you need to determine the least common multiple (LCM) of all the denominators involved. This means finding the smallest number that each of the denominators can divide into without leaving a remainder. In our exercise, we encountered the fractions \( \frac{x}{7} \) and \( \frac{5}{3} \). Here, our task was to find the LCM of 7 and 3, which is 21. A simple way to grasp this concept is:

• List the multiples of each number.
• The smallest shared number is the LCD.

By transforming the denominators to a common value, we simplify the task of combining fractions.

Combining Fractions

Once fractions have a common denominator, combining them becomes straightforward. This essentially means adding or subtracting the numerators while maintaining the common denominator. Consider the fractions with the least common denominator 21: \[ \frac{3x}{21} + \frac{35}{21} \]You simply add the numerators together, giving:\[ \frac{3x + 35}{21} \]Remember, the denominator stays the same because the parts we're combining are now comparable. This method not only simplifies the process but also ensures accuracy when working with algebraic expressions.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operational symbols. In our example, the expression \( \frac{x}{7} + \frac{5}{3} \) involves a variable \( x \), which signifies a quantity that may change. Understanding these expressions involves recognizing their components:

• Terms: Parts of the expression separated by addition or subtraction symbols.
• Coefficients: The numbers that multiply the variables.
• Variables: Letters that represent unknown values or can be any number.

Working with algebraic fractions means applying these concepts in conjunction to operations such as finding the LCD and combining the terms effectively. Recognizing how these pieces fit together enhances comprehension and problem-solving skills in algebra.