Problem 60 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{5 x+5 y}{7}+\frac{x^{2}-y^{2}}{x-y}$$

Step-by-Step Solution

Answer

\(\frac{12x+12y}{7}\)

Step 1: Simplify the fractions

Begin by simplifying each individual term of the problem. Notice that the whole expression can be divided into two main parts: \(\frac{5x+5y}{7}\) and \(\frac{x^2-y^2}{x-y}\). The second fraction is a special combination known as the difference of squares in the numerator which can be rewritten as \((x-y)(x+y)\). So the expression can be simplified to: \(\frac{5x+5y}{7} + (x+y)\)

Step 2: Make the denominator of fractions same

In order to add two fractions, you need to ensure that the denominators are the same. If the denominators are not the same, you will have to find a common denominator. In this particular case, the denominators are 7 and 1. So, the least common denominator is 7. We rewrite the expression to have a common denominator: \(\frac{5x+5y}{7} + \frac{7(x+y)}{7}\)

Step 3: Add Fractions

Now, we add the fractions together using the common denominator. Add the numerators together: \(\frac{5x+5y+7x+7y}{7}\). Now we simplify this by grouping like terms, resulting in: \(\frac{12x+12y}{7}\)

Step 4: Simplify The Fraction

The final step is to simplify the fraction as much as possible. Notice that the numerator and the denominator both have a common factor of 1. Therefore, the fraction can be simplified no further. The final solution is: \(\frac{12x+12y}{7}\)

Key Concepts

Difference of Squares

The difference of squares is a powerful tool in algebra for simplifying expressions, especially when one or more terms can be expanded or factored. In our exercise, we encounter a term \[(x^2 - y^2)\] which is called the difference of squares. Notice that this can be factored as \[(x-y)(x+y)\].

• The first term is the square of \(x\)
• The second term is the square of \(y\)

The product of these factors illustrates why it's called the "difference": \[(a^2 - b^2 = (a-b)(a+b))\].
In practical use, write complex terms as difference of squares when possible, to make further simplifications easier. Recognizing this pattern allows us to simplify fractions by canceling out common terms. Explore similar patterns in your exercises to streamline problem-solving.

Common Denominator

Adding fractions requires them to have the same denominator, called the "common denominator." This ensures the numerators can be combined directly. Consider two fractions such as \[\frac{a}{b} + \frac{c}{d}\] where \(b\) and \(d\) are different. To find a common denominator, search for the least common multiple (LCM) of the denominators. In our original exercise, we had: \[\frac{5x+5y}{7} + (x+y)\] which needed adjustment.

• The denominators are 7 and 1. This makes the LCM 7 since any number times 1 is itself.
• Express the term \(x+y\) as a fraction over the common denominator: \[\frac{7(x+y)}{7}\].

By converting all elements to share the same denominator, solving becomes straightforward. The numerators are then simply combined and processed further.

Simplifying Fractions

The final step when working with algebraic fractions often involves simplifying them. An expression is fully simplified when both the numerator and the denominator have no common factors other than 1. This step gives the cleanest and most concise answer possible.

In tackling the exercise, our simplified expression was: \[\frac{12x+12y}{7}\].

• First, note if there are any common factors. Here the common factors are part of the numerator only: 12. The denominator has none to simplify further.
• As dividing by the denominator did not yield common terms, the fraction is already as simple as it can be.

An important aspect of simplification is to check if coefficients or terms themselves can be factored out equally. Always look to simplify expressions to their simplest terms to make the results more manageable and better understood.