If `(5)/(x)=(y)/(10) and x-y=y`, what is the value of y+x?

Options

A. `5`
B. `10`
C. `15`
D. `25`

Answer

To solve the problem, we will follow these steps:

1. **Write down the equations**: We have two equations given:
\[ \frac{5}{x} = \frac{y}{10} \]
\[ x - y = y \]

2. **Rearrange the second equation**: From the second equation, we can express \(x\) in terms of \(y\):
\[ x - y = y \implies x = 2y \]

3. **Substitute \(x\) in the first equation**: Now we substitute \(x = 2y\) into the first equation:
\[ \frac{5}{2y} = \frac{y}{10} \]

4. **Cross-multiply to eliminate the fractions**: This gives us:
\[ 5 \cdot 10 = 2y \cdot y \]
Simplifying this, we get:
\[ 50 = 2y^2 \]

5. **Solve for \(y^2\)**: Dividing both sides by 2:
\[ y^2 = 25 \]

6. **Take the square root**: Taking the square root of both sides gives us:
\[ y = 5 \quad \text{or} \quad y = -5 \]

7. **Find corresponding values of \(x\)**:
- If \(y = 5\):
\[ x = 2y = 2 \times 5 = 10 \]
- If \(y = -5\):
\[ x = 2y = 2 \times (-5) = -10 \]

8. **Calculate \(y + x\)**:
- For \(y = 5\) and \(x = 10\):
\[ y + x = 5 + 10 = 15 \]
- For \(y = -5\) and \(x = -10\):
\[ y + x = -5 + (-10) = -15 \]

9. **Conclusion**: The possible values of \(y + x\) are 15 and -15. Since the question asks for the value of \(y + x\) and the options provided include 15, we conclude:
\[ \text{The value of } y + x = 15 \]