If `x + 3y = - 3x + y,` then `(x ^(2))/( y ^(2))` is equal to

Options

A. `1/8`
B. `1/2`
C. `1/4`
D. 4

Answer

To solve the equation \( x + 3y = -3x + y \) and find the value of \( \frac{x^2}{y^2} \), we can follow these steps:

### Step 1: Rearranging the Equation
Start with the given equation:
\[ x + 3y = -3x + y \]
Now, let's move all terms involving \( x \) to one side and all terms involving \( y \) to the other side:
\[ x + 3x = y - 3y \]
This simplifies to:
\[ 4x = -2y \]

### Step 2: Solving for \( \frac{x}{y} \)
Now, we can express \( x \) in terms of \( y \):
\[ x = -\frac{2}{4}y = -\frac{1}{2}y \]
Thus, we have:
\[ \frac{x}{y} = -\frac{1}{2} \]

### Step 3: Finding \( \frac{x^2}{y^2} \)
Now, we need to find \( \frac{x^2}{y^2} \):
\[ \frac{x^2}{y^2} = \left(\frac{x}{y}\right)^2 = \left(-\frac{1}{2}\right)^2 \]
Calculating this gives:
\[ \frac{x^2}{y^2} = \frac{1}{4} \]

### Final Answer
Thus, the value of \( \frac{x^2}{y^2} \) is:
\[ \frac{x^2}{y^2} = \frac{1}{4} \]

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