Problem 31 - Prealgebra

Question

Solve each equation by first finding the LCD for the fractions in the equation and then multiplying both sides of the equation by it.(Assume \(x\) is not 0 in Problems \(39-46\).) $$\frac{x}{3}+\frac{x}{6}=5$$

Step-by-Step Solution

Answer

The solution is \( x = 10 \).

Step 1: Identify the Fractions

The equation given is \( \frac{x}{3} + \frac{x}{6} = 5 \). Observe the fractions in the equation: \( \frac{x}{3} \) and \( \frac{x}{6} \).

Step 2: Find the Least Common Denominator (LCD)

The denominators of the fractions are 3 and 6. The least common multiple of these numbers is 6. Thus, the least common denominator (LCD) is 6.

Step 3: Multiply Both Sides by the LCD

Multiply every term in the equation by the LCD (6) to eliminate the fractions: \[ 6 \left( \frac{x}{3} \right) + 6 \left( \frac{x}{6} \right) = 6 \times 5 \] Simplify to get:\[ 2x + x = 30 \]

Step 4: Simplify the Equation

Combine like terms on the left side: \[ 2x + x = 3x \]The equation simplifies to:\[ 3x = 30 \]

Step 5: Solve for x

Divide both sides of the equation by 3 to solve for \( x \): \[ x = \frac{30}{3} = 10 \]

Key Concepts

Understanding the Least Common Denominator (LCD)

When working with fractions in equations, finding the least common denominator (LCD) is essential. The LCD is the smallest number that is a common denominator of the given fractions. It allows you to add, subtract, or combine fractions with different denominators effectively. In the given exercise, we see two fractions: \( \frac{x}{3} \) and \( \frac{x}{6} \). Here, our task is to find a common denominator to simplify the process of solving the equation.

To identify the LCD, start by examining the denominators, which are 3 and 6. Check for the smallest number that both denominators can divide into evenly. In this case, the number 6 satisfies this condition since both 3 and 6 can divide into it without leaving a remainder. Thus, 6 is our LCD.

Solving Equations with Fractions

Equations with fractions can sometimes appear daunting due to the presence of multiple denominators. However, converting them into a more manageable format can simplify the process. Once the least common denominator (LCD) is found, multiply every term in the equation by this LCD to eliminate the fractions from the equation. This step transforms the equation into a simpler, everyday form, making it easier to solve.

In the example equation, \( \frac{x}{3} + \frac{x}{6} = 5 \), we have determined the LCD to be 6. By multiplying each term by 6, the fractions are removed:

• \( 6 \times \frac{x}{3} = 2x \)
• \( 6 \times \frac{x}{6} = x \)
• \( 6 \times 5 = 30 \)

After this transformation, the equation becomes \( 2x + x = 30 \), which is much simpler to work with.

Combining Like Terms in Equations

Once the equation has been simplified and fractions are eliminated, the next step is to combine like terms. Like terms are terms in an equation that have the same variable raised to the same power. Combining them makes the equation concise and straightforward to solve.

In our example, after eliminating the fractions, we arrive at the equation \( 2x + x = 30 \). Both terms on the left side share the variable \( x \), which makes them like terms. By adding them together, \( 2x + x \) simplifies to \( 3x \). Now, the equation looks like \( 3x = 30 \).

Combining like terms is crucial because it reduces the complexity of the equation, allowing us to solve for the variable efficiently. In the last step of this process, we solve for \( x \) by dividing both sides of the equation by 3, leading to \( x = 10 \). This result brings the solution to completion.