For every pair (x, y) in the rectangular coordinate plane, `f : (x, y) to (x, -8x+3y)`. What is the set of points for which `f : (x, y) to (x, y)` ?

Options

A. The point (-4, 0)
B. The point (4, 12)
C. The set of points (x, y) that satisfy the equation x = 4y
D. The set of points (x, y) that satisfy the equation y = 4x

Explanation

To solve the problem, we need to find the set of points \((x, y)\) such that the function \(f\) defined by \(f(x, y) = (x, -8x + 3y)\) is equal to \((x, y)\). This means we need to solve the equation:

\[ f(x, y) = (x, y) \]

This translates to:

\[ (x, -8x + 3y) = (x, y) \]

From this, we can equate the corresponding components:

1. The x-coordinates:
\[ x = x \]
This equation is always true and does not provide any new information.

2. The y-coordinates:
\[ -8x + 3y = y \]

Now, we will solve the second equation for \(y\):

### Step 1: Rearranging the equation
Subtract \(y\) from both sides:
\[ -8x + 3y - y = 0 \]
This simplifies to:
\[ -8x + 2y = 0 \]

### Step 2: Isolating \(y\)
Now, we can isolate \(y\) by adding \(8x\) to both sides:
\[ 2y = 8x \]

### Step 3: Dividing by 2
To solve for \(y\), divide both sides by 2:
\[ y = 4x \]

### Conclusion
The set of points \((x, y)\) that satisfy the condition \(f(x, y) = (x, y)\) is given by the equation:
\[ y = 4x \]

This represents a line in the rectangular coordinate plane.