Problem 31 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{11 w+6}{3 u(w-9)}-\frac{11 w}{3 w(w-9)}\)

Step-by-Step Solution

Answer

The result is \(\frac{w(11w - 11u + 6)}{3uw(w-9)}\)."

Step 1: Common Denominator

Determine the common denominator between the two fractions. The denominators are \(3u(w-9)\) and \(3w(w-9)\). The common denominator is their product without repeated factors, which is \(3uw(w-9)\).

Step 2: Adjust Numerators

Adjust each fraction's numerator to reflect the common denominator.- For the first fraction, multiply the numerator \((11w+6)\) by \(w\) to get \((11w^2 + 6w)\).- For the second fraction, multiply the numerator \(11w\) by \(u\) to get \(11wu\).

Step 3: Form a Single Fraction

Combine the two fractions using their adjusted numerators over the common denominator:\[ \frac{11w^2 + 6w}{3uw(w-9)} - \frac{11wu}{3uw(w-9)} \]Combine the numerators to form a single fraction:\[ \frac{11w^2 + 6w - 11wu}{3uw(w-9)} \]

Step 4: Simplify the Numerator

Simplify the expression in the numerator, \(11w^2 + 6w - 11wu\).- Combine like terms: The terms \(11w^2\) and \(-11wu\) do not simplify together, but notice that \(6w\) is separate. Write it as: \[11w^2 - 11wu + 6w\]Factor out common terms here if possible.

Step 5: Factor and Simplify Additionaly

Check if any additional factoring can simplify the expression. Factor out the common factor \(w\) from \(11w^2 - 11wu + 6w\):\[w(11w - 11u + 6)\]. This is the greatest simplification since \(11w - 11u + 6\) is indivisible by the denominator's non-linear terms.

Step 6: Complete Simplified Fraction

Place the factored numerator over the common denominator:\[ \frac{w(11w - 11u + 6)}{3uw(w-9)} \]The fraction is now simplified with the numerator being fully factored.

Key Concepts

Fraction Subtraction

Subtracting fractions, especially those with algebraic expressions, involves a few clear steps. Firstly, each fraction is considered separately as having a distinct numerator and denominator, even though each may contain variables. When you're subtracting fractions, both must share the same denominator. This makes the subtraction process much like regular arithmetic subtraction, once both expressions are over the same denominators.

• Write both fractions with a common denominator.
• Subtract the numerators while keeping the denominator the same.

Approaching the subtraction of fractions with variables can at first appear daunting, but by methodically following the steps, the problem becomes more manageable.

Common Denominator

In order to subtract fractions, finding a common denominator is a critical step. This involves determining the least common multiple of the given denominators. For the algebraic fractions in this problem:

• The denominators are \(3u(w-9)\) and \(3w(w-9)\).
• The shared factors, like \((w-9)\), do not need to be repeated more than once.
• The least common denominator (LCD) is chosen to be \(3uw(w-9)\), which combines these without unnecessary repetition.

By converting each fraction to have this common denominator, subtraction is straightforward, since only the numerators need to be adjusted and subtracted. Remember, the LCD is essential in ensuring the resulting single expression is equivalent to each fraction's initial form.

Factoring Polynomials

Factoring plays an essential role in simplifying algebraic expressions. In our case, when given the expression \(11w^2 + 6w - 11wu\), the aim is to rewrite it in a simplified form by finding common factors.

• Identify any common terms in the expression. Here, observe that each part of the expression has a common factor "w".
• Factor out "w" to simplify: \[ w(11w - 11u + 6) \]

Factoring simplifies expressions and makes them easier to work with, especially when you need to further simplify fractions by canceling common terms with the denominator.

Simplifying Algebraic Fractions

After finding a common denominator and factoring the numerator, the final step is simplifying. This involves reducing the fraction to its simplest form. Once the numerator is factored:

• The expression is \( \frac{w(11w - 11u + 6)}{3uw(w-9)} \).
• Analyze if any terms in the denominator can cancel out with those in the numerator.
• In our example, since the denominator contains \(w\) and \(w\) from the numerator can be canceled, it simplifies calculation.

By canceling common terms, the fraction is reduced to its simplest form, making it easier to handle in further calculations.