Problem 12 - Elementary Algebra

Question

Solve each of the equations. $$\frac{h}{5}+\frac{h}{4}=2$$

Step-by-Step Solution

Answer

The solution is \( h = \frac{40}{9} \).

Step 1: Identify a common denominator

To solve the equation \( \frac{h}{5} + \frac{h}{4} = 2 \), we first need to find a common denominator for the fractions. The denominators are 5 and 4. The least common multiple of 5 and 4 is 20, so we will use 20 as the common denominator.

Step 2: Rewrite the fractions with the common denominator

Convert \( \frac{h}{5} \) and \( \frac{h}{4} \) using 20 as the denominator:- \( \frac{h}{5} = \frac{4h}{20} \) because \( h \times \frac{4}{4} = \frac{4h}{20} \).- \( \frac{h}{4} = \frac{5h}{20} \) because \( h \times \frac{5}{5} = \frac{5h}{20} \).

Step 3: Combine the fractions

Add the fractions together: \[\frac{4h}{20} + \frac{5h}{20} = \frac{9h}{20}\]

Step 4: Solve the equation

Set the equation to \( \frac{9h}{20} = 2 \). To clear the fraction, multiply both sides by 20: \[9h = 40\]

Step 5: Solve for h

Divide both sides by 9 to solve for \( h \): \[h = \frac{40}{9}\]

Key Concepts

Solving Linear Equations

When it comes to solving linear equations, the goal is to isolate the variable on one side of the equation. In our exercise, we are asked to solve for the variable \( h \) in the equation \( \frac{h}{5} + \frac{h}{4} = 2 \). Linear equations like this involve unknown quantities that we solve by using operations that maintain equality. The primary operations we employ include:

• Adding or subtracting the same value from both sides.
• Multiplying or dividing both sides by the same non-zero number.

Here, we first deal with the fractions by converting them to have a common denominator (which we will explain further in the next sections). Once that is achieved, the fractions can be combined into a single term, allowing us to easily transpose terms until \( h \) is isolated, leading to our solution.

Fractions and Common Denominators

Handling fractions is crucial in algebra, especially when adding or subtracting them. For the equation \( \frac{h}{5} + \frac{h}{4} = 2 \), the denominators, 5 and 4, need to be unified to properly combine the fractions. This involves finding a common denominator, and the most efficient choice is the least common multiple (LCM) of the denominators.

In this case, the LCM of 5 and 4 is 20. Adjust each fraction so that it has this common denominator:

• Multiply \( \frac{h}{5} \) by \( \frac{4}{4} \) to get \( \frac{4h}{20} \).
• Multiply \( \frac{h}{4} \) by \( \frac{5}{5} \) to get \( \frac{5h}{20} \).

With common denominators, fractions can be added: \( \frac{4h}{20} + \frac{5h}{20} = \frac{9h}{20} \). Mastering this technique helps in simplifying equations and is a valuable tool for more advanced algebraic concepts.

Step-by-Step Problem Solving

Approaching math problems in a step-by-step manner is beneficial in building strong problem-solving skills. The task of solving \( \frac{h}{5} + \frac{h}{4} = 2 \) exemplifies this method:

1. **Identify the Objective**: We need to solve for \( h \).
2. **Transform to Common Denominators**: This lets us add the fractions together smoothly.
3. **Combine and Simplify**: Combine the fractions to form \( \frac{9h}{20} \).
4. **Isolate the Variable**: To clear the fraction, multiply both sides by 20, resulting in \( 9h = 40 \).
5. **Solve for the Variable**: Divide both sides by 9, achieving \( h = \frac{40}{9} \).

By breaking down complex problems into smaller, manageable steps, students can tackle equations reliably and independently. This strategy reinforces understanding and provides a clear path to follow, reducing anxiety when faced with seemingly complex tasks.