Problem 13 - Algebra for College Students

Question

Solve each equation. \(\frac{h}{2}-\frac{h}{3}+\frac{h}{6}=1\)

Step-by-Step Solution

Answer

The solution is \( h = 3 \).

Step 1: Find a common denominator

Examine the denominators in the equation \( \frac{h}{2} - \frac{h}{3} + \frac{h}{6} = 1 \). The common denominator for 2, 3, and 6 is 6. Rewrite each fraction with 6 as the denominator: \( \frac{3h}{6} - \frac{2h}{6} + \frac{h}{6} = 1 \).

Step 2: Combine the fractions

Combine the fractions on the left side by adding and subtracting the numerators: \( \frac{3h - 2h + h}{6} = 1 \). This simplifies to \( \frac{2h}{6} = 1 \).

Step 3: Simplify the fraction

Further simplify \( \frac{2h}{6} \) by canceling the common factor of 2 in the numerator and denominator: \( \frac{h}{3} = 1 \).

Step 4: Solve for \( h \)

Since \( \frac{h}{3} = 1 \), multiply both sides by 3 to solve for \( h \): \( h = 3 \).

Key Concepts

Common Denominator

When dealing with rational equations, it's important to find a common denominator in order to combine or compare fractions effectively. A common denominator is a shared multiple of the denominators of all fractions involved in the equation.
If you look at the equation we are solving \( \frac{h}{2} - \frac{h}{3} + \frac{h}{6} = 1 \), you'll notice the denominators are 2, 3, and 6.
To find the least common denominator (LCD), consider the least common multiple (LCM) of these numbers. For our case, the LCM of 2, 3, and 6 is 6, since 6 is the smallest number that both 2, 3, and 6 will divide into evenly.
Using the common denominator:

• Rewrite \( \frac{h}{2} \) as \( \frac{3h}{6} \)
• Rewrite \( \frac{h}{3} \) as \( \frac{2h}{6} \)
• Keep \( \frac{h}{6} \) the same, since it already has 6 as the denominator

Having a common denominator helps combine fractions together, a vital step in rational equation solving.

Fraction Simplification

Once all fractions have the same denominator, you can move forward by combining them. This involves adding and subtracting the numerators while keeping the common denominator the same. This process can be seen in the simplified equation:
\( \frac{3h}{6} - \frac{2h}{6} + \frac{h}{6} \).
Combine the numerators to find that \( \frac{3h - 2h + h}{6} = \frac{2h}{6} \).

Now, simplify the newly formed fraction \( \frac{2h}{6} \). Fraction simplification involves reducing the fraction to its simplest form. To do this, find the greatest common factor of the numerator and the denominator and divide both by this number.
Here, the numerator is 2h and the denominator is 6. The greatest common factor of 2 and 6 is 2. Therefore,
\( \frac{2h}{6} \) simplifies to \( \frac{h}{3} \).
Simplifying fractions is a crucial step in solving rational equations because it reduces complexity and aids further calculations.

Linear Equations

After simplifying the fractions, the equation transforms into a simpler linear equation \( \frac{h}{3} = 1 \). Linear equations are equations where the highest power of the variable is one, making them straightforward to solve.
To isolate the variable, multiply both sides by the denominator of the fraction, effectively removing the fraction. In this case, multiply both sides by 3:
\( \frac{h}{3} \times 3 = 1 \times 3 \).
This simplifies to \( h = 3 \).

The process of solving a linear equation often involves:

• Eliminating fractions by multiplying through by the LCM of denominators
• Isolating the variable on one side of the equation

Understanding linear equations is fundamental, as they form the base for more complex algebraic manipulations and solutions.