Solve the following equations. 5. $$ \frac{1}{5 x}=\frac{3}{x}+\frac{1}{5} $$

Question

UpStudy AI · Accepted Answer

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:**

$$
x = -14
$$
$$ x = -14 $$

Solve the following equations.
5.

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