Problem 42 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{2 a-7}{a-9}+\frac{3 a+5}{a-9} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions: $$\frac{2a - 7}{a - 9} + \frac{3a + 5}{a - 9}$$ Answer: The sum of the given rational expressions is $$\frac{5a - 2}{a - 9}.$$

Step 1: Identify the given fractions and their common denominator

The given fractions are $$ \frac{2a - 7}{a - 9} \quad \text{and} \quad \frac{3a + 5}{a - 9} $$ The common denominator is \((a - 9)\).

Step 2: Add the numerators of the fractions

Now that we know the common denominator, we can add the numerators of the fractions: $$ (2a - 7) + (3a + 5) $$

Step 3: Simplify the sum of the numerators

We simplify the sum by combining like terms: $$ (2a + 3a) + (-7 + 5) = 5a - 2 $$

Step 4: Write the result as a single fraction

Finally, we write the result as a single fraction with the common denominator: $$ \frac{5a - 2}{a - 9} $$ So the sum of the given rational expressions is $$ \frac{2a - 7}{a - 9} + \frac{3a + 5}{a - 9} = \frac{5a - 2}{a - 9}. $$

Key Concepts

Adding Rational Expressions

Adding rational expressions is similar to adding ordinary fractions. In both cases, you need to bring the expressions to a common denominator before adding the numerators. This allows you to add the fractions directly without any issues stemming from different denominators. A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. So, when adding these expressions, it’s important to work step by step to ensure accuracy.

• First, identify the denominators of the given rational expressions.
• If they are already the same, you can move directly to adding the numerators together.
• If not, you'll need to find the least common denominator (LCD) to rewrite each fraction equivalently.

By consistently following these steps, rational expressions can be added systematically, just like whole numbers or fractions.

Common Denominator

A common denominator is crucial for adding and subtracting rational expressions. It refers to a shared denominator between two or more fractions, allowing the numerators to be directly added or subtracted. Without a common denominator, adding or subtracting fractions would be inaccurate and misleading.
In the given example, both rational expressions \( \frac{2a - 7}{a - 9} \) and \( \frac{3a + 5}{a - 9} \) already have a common denominator of \((a - 9)\). This simplifies the process because no additional work is needed to align the denominators. When the denominators are different, you'd need to:

• Find the least common multiple of the denominators.
• Re-write each fraction with the common denominator, adjusting the numerators accordingly to keep the expression's value the same.

This process ensures all expressions share the same base, enabling straightforward addition or subtraction of rational expressions.

Simplifying Expressions

Once the rational expressions have been added, the newly formed expression might not be in its simplest form. Simplifying the expression means reducing it as much as possible while keeping the value the same. This is similar to reducing fractions in basic arithmetic.

To simplify the final expression, follow these steps:

• Combine like terms from the numerators; for example, add coefficients of similar terms such as the \(a\) terms in \((2a + 3a)\).
• Simplify constants by performing addition or subtraction.
• Check if there are any common factors between the numerator and the denominator that can be canceled out. This helps in reducing the fraction to its simplest form.

In our example, after adding the numerators, \((2a + 3a) + (-7 + 5)\) simplifies to \(5a - 2\), resulting in the fraction \(\frac{5a - 2}{a - 9}\). Keeping expressions simplified aids in clearer understanding and further mathematical manipulation.