Problem 6 - Algebra 1: Concepts and Skills

Question

Subtract. Simplify your answer. $$ \frac{12 k}{k^{2}}-\frac{3 k+7}{k^{2}} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{9k - 7}{k^{2}} \)

Step 1: Combine the fractions

As the fractions \(\frac{12k}{k^{2}}\) and \(\frac{3k+7}{k^{2}}\) have the same denominator \(k^{2}\), they can be combined as follows: \(\frac{12k}{k^{2}} - \frac{3k + 7}{k^{2}} = \frac{12k - (3k+7)}{k^{2}}\)

Step 2: Simplify the numerator

Expand and simplify the numerator, \(12k - (3k+7) = 12k - 3k - 7 = 9k - 7\)

Step 3: Write the simplified fraction

The simplified fraction will therefore be \(\frac{9k - 7}{k^{2}}\)

Key Concepts

Fraction Subtraction

When it comes to subtracting fractions, the process is often simple if you remember the key rule: keep the denominators the same while you subtract the numerators. This is because only like, or identical, denominators allow for direct subtraction of the fractional parts. In the exercise given, both fractions

• \( \frac{12k}{k^2} \)
• \( \frac{3k+7}{k^2} \)

share the same denominator, \( k^2 \). This setup makes it easier to combine them by subtracting the numerators directly. Always subtract the entire second numerator from the first, paying close attention to distribute any negative signs across terms within brackets. Make sure your subtraction accounts for each term individually to avoid mistakes.

Like Denominators

The concept of like denominators is essential when working with fraction addition or subtraction. A like denominator means both fractions have the exact same bottom part, allowing direct manipulation of the top numbers, or numerators. In our problem, both fractions have \( k^2 \) as a denominator, so you can combine them directly by subtracting the numerators.
When denominators are identical, it's similar to dealing with whole numbers. For example, if you think about slices of pizza, combining fractions with like denominators is akin to combining pizza slices of the same size — six slices from one pizza minus three slices from another of the same size leaves you with three slices of the same pizza.
If the denominators are different, you'd normally need to find a common denominator first, which isn't the case here. This step simplifies your work, as you can focus directly on the numerators with confidence.

Simplifying Fractions

Simplifying fractions is about making the fraction as simple as possible. Once you perform subtraction, check if the resulting numerator and denominator can be reduced to simpler terms. In our example, after combining and simplifying the fractions, we are left with

• \( \frac{9k - 7}{k^2} \)

The key is to look for factors common to both the numerator and the denominator.
At this stage, you'd typically try to divide both if possible. However, in this case, \( 9k - 7 \) and \( k^2 \) have no common factors other than 1.
Thus, it's already in its simplest form. Always ensure that the expression is fully simplified before concluding your solution, as this makes the answer cleaner and sometimes easier to interpret. If a fraction reduces to an integer, this means the problem may have more elegant solutions or interpretations hidden in its simplicity.