If f(2x+3y,2x-7y)=20x, then f(x,y) equals

Options

A. x-y
B. 7x+3y
C. 3x-7y
D. None of these

Answer

To solve the problem, we need to find the function \( f(x, y) \) such that \( f(2x + 3y, 2x - 7y) = 20x \).

### Step-by-Step Solution:

1. **Identify the Variables**:
We have two variables in the function: \( u = 2x + 3y \) and \( v = 2x - 7y \). We need to express \( f(u, v) \) in terms of \( x \) and \( y \).

2. **Express \( x \) in terms of \( u \) and \( v \)**:
We can solve the equations for \( u \) and \( v \) to express \( x \) in terms of \( u \) and \( v \).
- From \( u = 2x + 3y \), we can express \( 2x = u - 3y \).
- From \( v = 2x - 7y \), we can express \( 2x = v + 7y \).

Setting these two expressions for \( 2x \) equal gives:
\[ u - 3y = v + 7y \]
Rearranging this, we find:
\[ u - v = 10y \quad \Rightarrow \quad y = \frac{u - v}{10} \]

3. **Substituting \( y \) back to find \( x \)**:
Now, substitute \( y \) back into one of the equations to find \( x \):
\[ 2x = u - 3\left(\frac{u - v}{10}\right) \]
Simplifying this:
\[ 2x = u - \frac{3u - 3v}{10} = u - \frac{3u}{10} + \frac{3v}{10} = \frac{10u - 3u + 3v}{10} = \frac{7u + 3v}{10} \]
Therefore:
\[ x = \frac{7u + 3v}{20} \]

4. **Finding \( f(u, v) \)**:
Given that \( f(2x + 3y, 2x - 7y) = 20x \), we substitute \( x \):
\[ f(u, v) = 20\left(\frac{7u + 3v}{20}\right) = 7u + 3v \]

5. **Final Expression**:
Thus, we have:
\[ f(x, y) = 7x + 3y \]

### Conclusion:
The function \( f(x, y) \) is \( 7x + 3y \).