Suppose f(x,y) is a real valued function such that f(3x+2y,2x−5y)=19x,forallrealnumbersxandy.Thevalueofxforwhichf(x, 2x) = 27 , is
TITA Answer:
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Official Correct Answer: 3. ### Core Concept
To solve the problem, we need to express the function \( f(a, b) \) in terms of \( a \) and \( b \). Given \( f(3x + 2y, 2x - 5y) = 19x \), we can solve for \( x \) in terms of \( a \) and \( b \) and substitute back into the function.
### Step-by-Step Solution
1. **Define Inputs**:
Let \( a = 3x + 2y \) and \( b = 2x - 5y \).
2. **Express \( x \) in Terms of \( a \) and \( b \)**:
- From the equations:
\[
a = 3x + 2y \quad \text{(1)}
\]
\[
b = 2x - 5y \quad \text{(2)}
\]
- Multiply equation (1) by 5: \( 15x + 10y = 5a \)
- Multiply equation (2) by 2: \( 4x - 10y = 2b \)
- Add the two equations:
\[
19x = 5a + 2b \implies x = \frac{5a + 2b}{19}
\]
3. **Express \( f(a, b) \)**:
- Substitute \( x \) into \( f(a, b) = 19x \):
\[
f(a, b) = 19 \times \frac{5a + 2b}{19} = 5a + 2b
\]
4. **Find \( x \) for \( f(x, 2x) = 27 \)**:
- Substitute \( a = x \) and \( b = 2x \) into \( f(a, b) \):
\[
f(x, 2x) = 5x + 2(2x) = 5x + 4x = 9x
\]
- Set equal to 27:
\[
9x = 27 \implies x = 3
\]
### 30-Second Shortcut
Recognize that \( f(a, b) \) is linear in \( a \) and \( b \). By solving the system of equations for \( x \) in terms of \( a \) and \( b \), we can directly substitute into \( f(a, b) \) and solve for \( x \).
### Final Answer
Correct Answer: 3