Question
Solve the following equations. 5. \( \frac{1}{5 x}=\frac{3}{x}+\frac{1}{5} \)
Ask by Dunn Chavez. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( x = -14 \)
Solution
To solve the equation:
\[
\frac{1}{5x} = \frac{3}{x} + \frac{1}{5}
\]
**Step 1: Eliminate the Denominators**
First, identify the least common denominator (LCD) of the fractions involved, which is \(5x\). Multiply every term in the equation by \(5x\) to eliminate the denominators:
\[
5x \cdot \frac{1}{5x} = 5x \cdot \frac{3}{x} + 5x \cdot \frac{1}{5}
\]
**Step 2: Simplify Each Term**
\[
1 = 15 + x
\]
**Step 3: Solve for \(x\)**
Subtract 15 from both sides to isolate \(x\):
\[
1 - 15 = x \\
-14 = x
\]
**Step 4: Verify the Solution**
Ensure that \(x = -14\) does not make any denominator zero:
- \(5x = 5(-14) = -70 \neq 0\)
- \(x = -14 \neq 0\)
Since neither denominator is zero, \(x = -14\) is a valid solution.
**Final Answer:**
\[
x = -14
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To solve the equation \( \frac{1}{5 x} = \frac{3}{x} + \frac{1}{5} \), we first find a common denominator, which is \( 5x \). This allows us to rewrite the equation as: \[ \frac{1}{5x} = \frac{15}{5x} + \frac{x}{5x} \] Now, we combine the terms on the right side: \[ \frac{1}{5x} = \frac{15 + x}{5x} \] Since the denominators are the same, we can equate the numerators: \[ 1 = 15 + x \] To isolate \( x \), subtract 15 from both sides: \[ x = 1 - 15 \] Thus, \[ x = -14 \] The solution to the equation is \( x = -14 \).
