If `((x)/(y))^(5a-3)=((y)/(x))^(17-3a)`, what is the value of a ?

Options

A. `-7`
B. `-5`
C. 0
D. 3

Answer

To solve the equation \(\left(\frac{x}{y}\right)^{5a-3} = \left(\frac{y}{x}\right)^{17-3a}\), we can follow these steps:

### Step 1: Rewrite the right side of the equation
We know that \(\frac{y}{x} = \left(\frac{x}{y}\right)^{-1}\). Therefore, we can rewrite the right side:
\[ \left(\frac{y}{x}\right)^{17-3a} = \left(\left(\frac{x}{y}\right)^{-1}\right)^{17-3a} = \left(\frac{x}{y}\right)^{-(17-3a)} \]
So, the equation becomes:
\[ \left(\frac{x}{y}\right)^{5a-3} = \left(\frac{x}{y}\right)^{-(17-3a)} \]

### Step 2: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ 5a - 3 = -(17 - 3a) \]

### Step 3: Simplify the equation
Distributing the negative sign on the right side gives:
\[ 5a - 3 = -17 + 3a \]

### Step 4: Rearrange the equation
To isolate \(a\), we can move \(3a\) to the left side and \(-3\) to the right side:
\[ 5a - 3a = -17 + 3 \]
This simplifies to:
\[ 2a = -14 \]

### Step 5: Solve for \(a\)
Dividing both sides by 2 gives:
\[ a = -7 \]

### Final Answer
Thus, the value of \(a\) is:
\[ \boxed{-7} \]