Problem 41 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{8 x}{3}-\frac{3 x}{7}$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{47x}{21} \).

Step 1: Identify the Objective

The task is to subtract two rational expressions: \( \frac{8x}{3} - \frac{3x}{7} \). Our goal is to express the result in its simplest form, which involves finding a common denominator, subtracting the numerators, and simplifying the expression.

Step 2: Find the Common Denominator

To subtract rational expressions, we need a common denominator. The denominators here are 3 and 7. The least common denominator (LCD) of 3 and 7 is 21.

Step 3: Rewrite Fractions with Common Denominator

Adjust each fraction so that they both have the denominator of 21:\[ \frac{8x}{3} = \frac{8x \cdot 7}{3 \cdot 7} = \frac{56x}{21} \]\[ \frac{3x}{7} = \frac{3x \cdot 3}{7 \cdot 3} = \frac{9x}{21} \]

Step 4: Subtract the Numerators

With a common denominator, subtract the numerators:\[ \frac{56x}{21} - \frac{9x}{21} = \frac{56x - 9x}{21} = \frac{47x}{21} \]

Step 5: Simplify the Expression

Check if the fraction \( \frac{47x}{21} \) can be further simplified. Since 47 and 21 have no common factors (besides 1), the expression is already in its simplest form.

Key Concepts

Common Denominator

When dealing with the addition or subtraction of algebraic fractions, finding a common denominator is essential. Much like adding or subtracting ordinary fractions, algebraic fractions must have the same denominator to be combined.
The common denominator for algebraic fractions is essentially the least common multiple (LCM) of the individual denominators. In our example, the denominators are 3 and 7.
Finding the LCM of two numbers involves identifying the smallest number that both denominators divide into evenly.

• For numbers like 3 and 7, which are primes, the LCM is simply their product: 3 times 7 equals 21.

Once the common denominator is determined, each fraction is adjusted to have this denominator by appropriate multiplication in both the numerator and the denominator. This step is crucial as it lays the groundwork for correctly performing addition or subtraction amongst fractions.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. This involves reducing the numerator and the denominator to their smallest integer values such that they have no common factors, other than 1.
In our problem, after subtraction, we obtained the fraction \( \frac{47x}{21} \). We then check to see if these numbers share any common factors that could further reduce the fraction.

• Check each number starting with the smallest prime number (in this case 2), and move upwards (3, 5, 7,...).
• If no factors emerge by these smaller primes, attempt only the prime numbers up to the square root of the smaller number involved in the division.
• Here, since 47 is prime, and 21 is not divisible by 47, the fraction \( \frac{47x}{21} \) is already in its simplest form.

By simplifying fractions, we ensure that our solution is as clear and concise as possible, which is especially useful when further calculations may be needed.

Rational Expressions

Rational expressions are like fractions but they contain polynomials in their numerators and denominators. They require special consideration during operations such as addition, subtraction, multiplication, and division.
Just like regular fractions, when working with rational expressions, the goal often includes simplifying or putting them into a more manageable form.

• Identify the expression: In our task, both fractions, \( \frac{8x}{3} \) and \( \frac{3x}{7} \), are rational expressions since they include variables in their numerators.
• Perform operations: Depending on the choice of operation, ensure similarity in denominators before proceeding with numerators like regular fractions.
• Simplify the result: Following operations like subtraction, check if the resulting expression can be simplified further to its simplest terms.

These steps maintain the integrity of the expressions while allowing algebraic manipulation to solve the equations simpler, making solving equations with variables more efficient.