Problem 70 - Algebra for College Students

Question

Either solve the given equation or perform the indicated operation \((s),\) whichever is appropriate. $$\frac{5}{h}-\frac{h}{5}$$

Step-by-Step Solution

Answer

\frac{25 - h^2}{5h}

Step 1: Identify a Common Denominator

To subtract the fractions \(\frac{5}{h}-\frac{h}{5}\), find a common denominator for \(h\) and \(5\). The common denominator is the product \(h \times 5 = 5h\).

Step 2: Rewrite Each Fraction

Express each fraction with the common denominator \(5h\). For \(\frac{5}{h}\), multiply the numerator and denominator by 5. It becomes \(\frac{5 \times 5}{h \times 5} = \frac{25}{5h}\). For \(\frac{h}{5}\), multiply the numerator and denominator by \(h\). It becomes \(\frac{h \times h}{5 \times h} = \frac{h^2}{5h}\).

Step 3: Subtract the Fractions

Subtract the fractions by keeping the common denominator and combining the numerators: \(\frac{25}{5h} - \frac{h^2}{5h} = \frac{25 - h^2}{5h}\).

Key Concepts

Common Denominator

When dealing with the subtraction (or addition) of fractions, a common denominator is essential. The common denominator allows us to combine the fractions into one. Imagine trying to subtract 1/2 from 1/3 directly—because they have different denominators, it's like subtracting apples from oranges. We need to

Fraction Subtraction

Once we have a common denominator, we can subtract fractions easily. Let's use the example from the exercise: \( \frac{5}{h} - \frac{h}{5}\). After finding our common denominator, which is \(5h\), we rewrite the fractions.
So, the fractions become \( \frac{25}{5h} \) and \( \frac{h^{2}}{5h} \). Now we can subtract them because they have the same denominator! So, \( \frac{25}{5h} - \frac{h^2}{5h} = \frac{25 - h^2}{5h} \).
After the fractions are rewritten, always keep the common denominator, and focus on subtracting the numerators.
Always ensure that the common denominator remains unchanged because it unifies the fractions into a single framework.

Algebraic Manipulation

The final step in our problem is algebraic manipulation, which involves simplifying the expression.
By using algebra, we rearrange, combine, and simplify expressions to make them easier to work with. In the given step where we have \( \frac{25 - h^2}{5h} \), we started by ensuring both fractions shared a common denominator.
Combining these fractions and then focusing on the numerator is essential.
Algebraic manipulation often consolidates multiple steps into one simplified form.