Problem 38 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{9 x}{x-10}-\frac{x}{x-3}\)

Step-by-Step Solution

Answer

\(\frac{8x^2 - 17x}{(x-10)(x-3)}\) is the simplified expression.

Step 1: Identify the Least Common Denominator (LCD)

The given expression is \( \frac{9x}{x-10} - \frac{x}{x-3} \). To combine these fractions, we first find the least common denominator. The denominators \(x-10\) and \(x-3\) have no common factors, so their LCD is \((x-10)(x-3)\).

Step 2: Rewrite Each Fraction with the LCD

Rewrite each fraction so that both have the common denominator \((x-10)(x-3)\). For \( \frac{9x}{x-10} \), multiply the numerator and denominator by \(x-3\):\[\frac{9x(x-3)}{(x-10)(x-3)} = \frac{9x^2 - 27x}{(x-10)(x-3)}\]For \( \frac{x}{x-3} \), multiply the numerator and denominator by \(x-10\):\[\frac{x(x-10)}{(x-3)(x-10)} = \frac{x^2 - 10x}{(x-3)(x-10)}\]

Step 3: Subtract the Numerators

Now that the fractions share a common denominator, subtract the numerators:\[\frac{9x^2 - 27x}{(x-10)(x-3)} - \frac{x^2 - 10x}{(x-3)(x-10)}\]This simplifies to:\[\frac{(9x^2 - 27x) - (x^2 - 10x)}{(x-10)(x-3)}\]Distribute and combine like terms:\[\frac{9x^2 - 27x - x^2 + 10x}{(x-10)(x-3)} = \frac{8x^2 - 17x}{(x-10)(x-3)}\]

Step 4: Simplify the Expression

The expression \(\frac{8x^2 - 17x}{(x-10)(x-3)}\) cannot be factored further in a way that allows for cancellation with the denominator, so this is the simplified form of the expression.

Key Concepts

Least Common Denominator

When dealing with fractions, establishing a common denominator is crucial, especially if the denominators aren't the same. The least common denominator (LCD) is simply the smallest expression that both denominators can divide into without leaving a remainder.

Let's take a closer look at our problem: \[ rac{9x}{x-10} - rac{x}{x-3}. \] Here, the denominators are \(x-10\) and \(x-3\). These are distinct terms, meaning they don't share any factors. Thus, the least common denominator is simply their product: \[(x-10)(x-3).\]

This product serves as the common base for both fractions, allowing us to combine them later. By recognizing the LCD upfront, we pave the way for smooth calculations and proper manipulation of the expressions.

Simplifying Expressions

Simplifying expressions involves reducing them into their most compact form. After finding the LCD, each fraction needs to be rewritten to reflect this new common base.

For the fraction \( \frac{9x}{x-10}, \) we multiply the numerator and denominator by \( x-3, \) resulting in:\[ \frac{9x(x-3)}{(x-10)(x-3)} = \frac{9x^2 - 27x}{(x-10)(x-3)}. \]Similarly, for \( \frac{x}{x-3}, \) we multiply by \( x-10: \)\[ \frac{x(x-10)}{(x-3)(x-10)} = \frac{x^2 - 10x}{(x-3)(x-10)}. \]

These operations ensure each fraction shares the same denominator, setting the stage for subtraction. This step is crucial, as it allows us to work with numerators that can now be manipulated directly, thanks to a unified denominator.

Subtracting Fractions

Once we've made sure both fractions have the same denominator, subtraction becomes straightforward. Here's a simple way to handle this:

• Subtract the numerators: Instead of subtracting entire fractions, focus solely on their numerators. For our expressions: \[ \frac{9x^2 - 27x}{(x-10)(x-3)} - \frac{x^2 - 10x}{(x-3)(x-10)} \]we subtract \( (x^2 - 10x) \) from \( (9x^2 - 27x). \)
• Simplify the result: After carrying out the subtraction, simplify the result as much as possible. In our case, this becomes:\[ \frac{(9x^2 - 27x) - (x^2 - 10x)}{(x-10)(x-3)} = \frac{8x^2 - 17x}{(x-10)(x-3)}. \]

The key to mastering this process lies in careful manipulation of numerators, ensuring no steps are skipped. After subtracting, always check if the expression can be further simplified. In our example, the numerator is as simple as it can get, so the provided answer is already optimal.