Problem 4 - Elementary Algebra

Question

Add or Subtract the following rational expressions. $$ \frac{x+2}{x-y}+\frac{2 x+3 u}{x-y} $$

Step-by-Step Solution

Answer

Question: Add the rational expressions and simplify: $$ \frac{x+2}{x-y}+\frac{2x+3u}{x-y} $$ Answer: The simplified rational expression is: $$ \frac{3x+2+3u}{x-y} $$

Step 1: Identify the common denominator

Here, both fractions have the same denominator, which is \((x-y)\).

Step 2: Add the numerators

We add the numerators of the two fractions while keeping the common denominator: $$ \frac{x+2}{x-y}+\frac{2x+3u}{x-y} = \frac{(x+2)+(2x+3u)}{x-y} $$

Step 3: Simplify the numerator

Combine the like terms in the numerator: $$ \frac{(x+2)+(2x+3u)}{x-y} = \frac{1x+2x+2+3u}{x-y} = \frac{3x+2+3u}{x-y} $$

Step 4: Write the final answer

The simplified rational expression is: $$ \frac{3x+2+3u}{x-y} $$

Key Concepts

Common Denominator

Understanding how to find a common denominator is critical when dealing with the addition of rational expressions. The denominator serves as the common ground that allows for the combination of the numerators. In the given exercise, the process begins with spotting that the two fractions \( \frac{x+2}{x-y} \) and \( \frac{2x+3u}{x-y} \) share the same denominator, \( x-y \). This is a huge relief because it simplifies the process of addition.

When fractions have different denominators, we must find the least common denominator (LCD). The LCD is the smallest expression that both denominators can divide into evenly. It may be the product of the denominators or a smaller expression if they share common factors. Once the LCD is determined, both fractions are converted to equivalent fractions with the newly found common denominator before they can be added or subtracted.

Simplifying Numerators

The next step, simplifying numerators, is where we look to combine like terms and reduce the complexity of our numerical expression. In our exercise, once we've established a common denominator, we focus on the numerators. We sum them up directly: \(x+2+2x+3u\). This is where students might stumble if they do not remember to combine only the like terms.

After the initial sum, we identify and group like terms, which in this case are the terms containing 'x'. Thus, we get \(1x+2x+2+3u\), which simplifies to \(3x+2+3u\) after combining. By simplifying the numerator in this manner, not only our expression becomes neater, but it also becomes more understandable and, hence, more functional for further mathematical operations.

Algebraic Fractions

Dealing with algebraic fractions can often be intimidating because they involve not just numbers, but also variables. These fractions are just like traditional fractions, but their numerators, denominators, or both contain algebraic expressions. The key when working with algebraic fractions is to treat the variables as if they were numbers, following the same rules for addition, subtraction, multiplication, and division.

When adding or subtracting algebraic fractions, we first check to ensure that the denominators are the same, just as we would with numerical fractions. If not, we find a common denominator. The same principles of combining like terms and simplifying apply. It is essential to be cautious and avoid the mistake of canceling out terms across the numerator and denominator unless they are factors of the entire expression.

Rational Expression Addition

When it comes to rational expression addition, we are essentially combining two or more algebraic fractions. The process involves steps similar to adding numerical fractions but with the additional complexity of dealing with variables.

First, ensure a common denominator is in place. Next, add the numerators, which might involve combining like terms. The final step is to simplify the resulting expression. It's crucial to present the answer in its simplest form to avoid any unnecessary complexity. In our example, the addition of the rational expressions \(\frac{x+2}{x-y}\) and \(\frac{2x+3u}{x-y}\) leads to a single, simplified expression \(\frac{3x+2+3u}{x-y}\), which is the final and simplest form of the rational expression after addition.