Problem 33 - 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 \(39-46\).) $$\frac{x}{5}-x=4$$

Step-by-Step Solution

Answer

The solution is \( x = -5 \).

Step 1: Identify the Fractions

Identify the fractions in the equation. Here, the equation is \( \frac{x}{5} - x = 4 \). The fraction present is \( \frac{x}{5} \).

Step 2: Determine the Least Common Denominator (LCD)

Since there's only one fraction with a denominator of 5, the LCD for this equation is 5.

Step 3: Multiply the Entire Equation by the LCD

Multiply every term in the equation by the LCD, which is 5, to eliminate the fraction. The equation becomes: \[ 5 \cdot \frac{x}{5} - 5 \cdot x = 5 \cdot 4 \] Simplifying each term gives us: \[ x - 5x = 20 \]

Step 4: Simplify the Equation

Combine the terms on the left-hand side of the equation: \[ -4x = 20 \]

Step 5: Solve for x

Divide both sides by -4 to find the value of \( x \): \[ x = \frac{20}{-4} \] Simplifying gives: \[ x = -5 \]

Key Concepts

Understanding the Least Common Denominator (LCD)

In equations with fractions, combining terms can be tricky. One useful technique is finding the Least Common Denominator (LCD). The LCD is the smallest number that all denominators can divide into evenly. When you find the LCD, you can remove fractions from an equation, making it much simpler to solve.

• To find the LCD, look at the denominators in your equation.
• In our example, the equation is \(\frac{x}{5} - x = 4\). The only fraction is \(\frac{x}{5}\) with a denominator of 5.
• Given only one fraction, the denominator itself is the LCD, namely 5.

Using the LCD lets you clear fractions by multiplying every term by this number. It paves the way for an easier solution process, transforming tricky fractions into straightforward algebraic terms.

Fraction Elimination Technique

Once we've found the LCD, the next step is to eliminate fractions from the equation. This technique involves multiplying each term in the equation by the LCD. Doing this gets rid of any fractions completely. In our example:

• The equation began as \(\frac{x}{5} - x = 4\).
• By multiplying every term by the LCD (5), it becomes: \[ 5 \cdot \frac{x}{5} - 5 \cdot x = 5 \cdot 4 \]
• Simplifying gives us: \[ x - 5x = 20 \]

Notice how the fraction is no more! This tactic is powerful because it turns complex equations into simpler ones, making it straightforward to find solutions.

Simplifying the Equation

After eliminating fractions, the focus shifts to simplifying the equation. This part is about cleaning up and solving the resulting expression. For our problem:

• Once the equation is transformed to \( x - 5x = 20 \), we address any combined like terms.
• The terms \( x - 5x \) simplify to \( -4x \).
• This leaves us with \( -4x = 20 \).
• To isolate \( x \), divide both sides by -4, resulting in: \[ x = \frac{20}{-4} \]
• Finally, simplify the fraction to find \( x = -5 \).

Equation simplification is a systematic approach. It's about combining similar terms and isolating the variable to solve it successfully, shedding any complexities at each step.