If `(x + 5) / (1/x + 1/5)= 5`, then x =

Options

A. `-5`
B. `1//2`
C. `1`
D. `5`

Answer

To solve the equation \((x + 5) / (1/x + 1/5) = 5\), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the equation:
\[ \frac{x + 5}{\frac{1}{x} + \frac{1}{5}} = 5 \]

### Step 2: Simplify the Denominator
To simplify the denominator \(\frac{1}{x} + \frac{1}{5}\), we need to find a common denominator:
\[ \frac{1}{x} + \frac{1}{5} = \frac{5 + x}{5x} \]
So, we can rewrite the equation as:
\[ \frac{x + 5}{\frac{5 + x}{5x}} = 5 \]

### Step 3: Multiply by the Denominator
Now, we multiply both sides by \(\frac{5 + x}{5x}\) to eliminate the fraction:
\[ (x + 5) \cdot \frac{5x}{5 + x} = 5 \]
This simplifies to:
\[ 5x(x + 5) = 5(5 + x) \]

### Step 4: Expand Both Sides
Expanding both sides gives:
\[ 5x^2 + 25x = 25 + 5x \]

### Step 5: Rearrange the Equation
Now, we rearrange the equation to bring all terms to one side:
\[ 5x^2 + 25x - 5x - 25 = 0 \]
This simplifies to:
\[ 5x^2 + 20x - 25 = 0 \]

### Step 6: Divide by 5
To simplify, we can divide the entire equation by 5:
\[ x^2 + 4x - 5 = 0 \]

### Step 7: Factor the Quadratic
Next, we factor the quadratic equation:
\[ (x + 5)(x - 1) = 0 \]

### Step 8: Solve for x
Setting each factor to zero gives us:
\[ x + 5 = 0 \quad \text{or} \quad x - 1 = 0 \]
Thus, we find:
\[ x = -5 \quad \text{or} \quad x = 1 \]

### Conclusion
The solution to the equation is:
\[ x = 1 \quad \text{(since we need a valid solution in the context of the original equation)} \]