Problem 53 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{c}{7 c-d}-\frac{d}{d-7 c} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{c + d}{7c-d}\).

Step 1: Recognize Identical Denominators

Observe that the two denominators, \(7c - d\) and \(d - 7c\), are opposites of each other. Specifically, \(d - 7c = -(7c - d)\). This means they will simplify upon combining the fractions.

Step 2: Rewrite the Second Fraction

Convert the second fraction, \(-\frac{d}{d-7c}\), by recognizing that \(d - 7c = -(7c - d)\). Thus, we can rewrite the second fraction as \(\frac{d}{-(7c - d)} = \frac{-d}{7c-d}\).

Step 3: Combine the Fractions

Now that both fractions have the same denominator, \(7c-d\), subtract the second fraction from the first: \(\frac{c}{7c-d} - \frac{-d}{7c-d} = \frac{c + d}{7c-d}\).

Step 4: Simplify the Expression

Check to see if any further simplifications can be made with the expression \(\frac{c + d}{7c-d}\). In this case, \(c+d\) and \(7c-d\) do not share any common factors, so the expression is already in its simplest form.

Key Concepts

Understanding Opposite Denominators

When dealing with algebraic fractions, identifying opposite denominators is key. Opposite denominators appear when you have two expressions in the denominator that are negative inverses of each other. For example, the denominators \(7c - d\) and \(d - 7c\) are opposites. We can express this relationship algebraically: \(d - 7c = -(7c - d)\).
Understanding this concept helps simplify the process of combining these fractions because recognizing opposite denominators allows you to manipulate fractions into a common form.

• This is crucial for combining fractions effectively.
• Knowing how to identify and handle opposite denominators saves you time in algebraic manipulation.

Once you recognize opposite denominators, you can rewrite fractions and proceed with other operations such as simplification or subtraction.

The Art of Fraction Simplification

Fraction simplification in algebra is all about reducing expressions to their simplest form while ensuring equivalent values. When you simplify a fraction, you look to cancel common factors in the numerator and the denominator. For instance, in the expression \(\frac{c + d}{7c-d}\), check the numerator and the denominator for common factors.
In this case, both parts, \(c + d\) and \(7c - d\), have no common factors, which means the fraction is already simplified. A fraction is fully simplified when:

• Numerator and denominator have no common factors besides 1.
• You can't divide both by the same variable or constant to get simpler numbers.

Simplifying fractions is part of making algebraic expressions more manageable and is a step toward solving polynomial equations accurately.

Combining Fractions with Confidence

Combining fractions is a fundamental skill in algebra that allows you to add or subtract fractions with different parts effectively. Before combining fractions, the denominators must be the same, which might involve rewriting one or both fractions.
Once you standardize the denominators, as shown when we rewrote \(-\frac{d}{d-7c}\) to \(\frac{-d}{7c-d}\), combining them becomes straightforward: - Subtract the numerators: \( \frac{c}{7c-d} - \frac{-d}{7c-d} \)- Calculate: \( \frac{c + d}{7c-d} \)
The key processes involve:

• Changing opposite denominators to a common denominator.
• Adding or subtracting numerators directly.

Mastery of combining fractions poises students to solve more complex equations where fractions play a role. Follow these steps and strategies to increase your fluency in working with algebraic fractions.