Problem 41 - College Algebra

Question

Add and subtract the rational expressions, and then simplify. $$ \frac{x}{x+1}+\frac{y}{y+1} $$

Step-by-Step Solution

Answer

\(\frac{2xy + x + y}{(x + 1)(y + 1)}\) is the simplified expression.

Step 1: Identify Common Denominator

To add the rational expressions, first identify a common denominator. In this case, the common denominator will be the product of the two separate denominators: \[(x + 1)(y + 1)\].

Step 2: Rewrite Each Expression with the Common Denominator

Rewrite each rational expression to have the common denominator:For \(\frac{x}{x+1}\), multiply the numerator and denominator by \((y+1)\): \[\frac{x(y+1)}{(x+1)(y+1)}\]For \(\frac{y}{y+1}\), multiply the numerator and denominator by \((x+1)\):\[\frac{y(x+1)}{(x+1)(y+1)}\]

Step 3: Add the Numerators

Add the numerators of the rewritten expressions over the common denominator:\[\frac{x(y+1) + y(x+1)}{(x+1)(y+1)}\]

Step 4: Simplify the Numerator

Simplify the numerator by expanding and combining like terms:\[x(y+1) + y(x+1) = xy + x + yx + y = 2xy + x + y\]Now the expression becomes:\[\frac{2xy + x + y}{(x+1)(y+1)}\]

Key Concepts

Common Denominator

The concept of a common denominator is essential when working with rational expressions, especially for addition or subtraction. A rational expression is similar to a fraction, where the numerator and the denominator can be polynomials. To add or subtract these expressions, they must have the same denominator. This is called the common denominator.

• First, identify the denominators of each rational expression you are working with.
• For the exercise here, the denominators are \(x+1\) and \(y+1\).
• The common denominator is obtained by multiplying these denominators together: \((x+1)(y+1)\).

This step ensures that you can combine the two expressions into a single fraction because they are now "speaking the same language" through a unified denominator.

Addition of Fractions

Once you have identified the common denominator, the next step involves rewriting each fraction. Rewriting is crucial so that each rational expression has the new common denominator. This often involves multiplying both the numerator and the denominator by factors that will make each denominator equivalent to the common one.

• For \(\frac{x}{x+1}\), multiply both top and bottom by \(y+1\), becoming \(\frac{x(y+1)}{(x+1)(y+1)}\).
• For \(\frac{y}{y+1}\), multiply both top and bottom by \(x+1\), becoming \(\frac{y(x+1)}{(x+1)(y+1)}\).

Now, you can add the numerators while keeping the common denominator the same. The result is a new rational expression that combines the initial ones into a simplified form.

Simplifying Algebraic Expressions

Simplifying the algebraic expression involves combining like terms and reducing the expression to its most straightforward form. Once the fractions are added, your expression might look complex. At this stage:

• Expand the expressions by distributing any multiplication inside the numerator.
• For the exercise, expand \(x(y+1) + y(x+1)\) to get \(xy + x + yx + y\).
• Combine like terms, in this case, \(xy\) and \(yx\), leading to \(2xy + x + y\).

Now the expression \(\frac{2xy + x + y}{(x+1)(y+1)}\) is in simplest form. Simplification is a vital skill where practice makes the concepts easier to understand and apply, allowing the expression to be manageable and more straightforward.