Problem 36 - 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 \(39-46\).) $$3 x-\frac{1}{3}=\frac{1}{6}$$

Step-by-Step Solution

Answer

The solution is \( x = \frac{1}{6} \).

Step 1: Identify the denominators

The given equation is \( 3x - \frac{1}{3} = \frac{1}{6} \). Here, the denominators present in the fractions are 3 and 6.

Step 2: Find the least common denominator (LCD)

To eliminate the fractions, we need to find the least common denominator of 3 and 6. The LCD of 3 and 6 is 6.

Step 3: Multiply each term by the LCD

Multiply every term in the equation by the LCD (which is 6): \( 6(3x) - 6(\frac{1}{3}) = 6(\frac{1}{6}) \).Simplifying this gives us: \( 18x - 2 = 1 \).

Step 4: Simplify and solve for x

Now, solve the equation \( 18x - 2 = 1 \). First, add 2 to both sides to isolate the term with \( x \):\( 18x = 1 + 2 \).This simplifies to:\( 18x = 3 \).Next, divide both sides by 18:\( x = \frac{3}{18} \).Simplifying \( \frac{3}{18} \) gives \( x = \frac{1}{6} \).

Key Concepts

Least Common Denominator

When dealing with equations that include fractions, one common approach is to eliminate these fractions by employing the concept of the Least Common Denominator (LCD). The LCD is the smallest number that all the denominators can divide into evenly.
For example, in the equation \(3x - \frac{1}{3} = \frac{1}{6}\), the denominators present are 3 and 6. You need to find the smallest number that is a multiple of both 3 and 6. In this case, the LCD is 6.
Finding the LCD is essential because it allows you to multiply through the entire equation to clear the fractions, making the equation easier to solve. By applying the LCD to each term in the equation, you allow for a straightforward simplification process.

Fractions

Fractions can sometimes make solving equations look intimidating, but they're just a way to express part of a whole. A fraction is made up of a numerator, which is the top number, and a denominator, which is the bottom number.
In the context of equations, fractions often arise as coefficients or constants that require careful handling. Take the example equation \(3x - \frac{1}{3} = \frac{1}{6}\). Here, \(\frac{1}{3}\) and \(\frac{1}{6}\) are the fractions involved. The denominators indicate the division, while the numerators show how many parts we have.

• By using the LCD, we convert these fractions into whole numbers. Multiplying by the LCD effectively scales up each fraction so that the denominators cancel out.

This transformation paves the way for solving the equation without the hassle of dealing with fractions.

Simplifying Equations

Simplifying equations involves using mathematical operations to rewrite it in a simpler and clearer form, making it easier to solve. After clearing fractions using the LCD, equations can usually be simplified more easily.
Continuing with our example, after multiplying each term by the LCD 6, you get the equation \(18x - 2 = 1\). From here, you need to isolate the variable \(x\) to solve for it. Start by adding 2 to both sides to undo the subtraction, resulting in \(18x = 3\).

• Next, divide both sides by 18 to solve for \(x\), giving: \(x = \frac{3}{18}\).
• Lastly, simplify \(\frac{3}{18}\) to \(\frac{1}{6}\), by dividing the numerator and the denominator by their greatest common factor.

Simplification is key in mathematics as it reveals the most direct and simple form of an answer, making it easy to understand and verify.