To reflect the line $y = 2x - 3$ across another line $y = x + 4$, we need to go through a series of steps. Here is a detailed step-by-step solution for this transformation:

Steps for Reflection Across a Line

1. Step 1: Parameterize the first line equation $y = 2x - 3$.

   • This is the original line that we want to reflect.
   • In parametric form, each point $P(x, y)$ on the line satisfies the equation $y = 2x - 3$.

2. Step 2: Find the general formula for reflecting a point.

   • To reflect a point $P(x_1, y_1)$ across a line $y = mx + b$ (in this case, $y = x + 4$), we can use the following procedure:
     • We find the perpendicular distance between the point and the reflection line.
     • Then, we calculate the reflected point by applying reflection formulas.

3. Step 3: Use the formula for reflecting a line across $y = x + 4$.

   • To reflect across the line $y = x + 4$, we first translate the line by subtracting 4 from all $y$-coordinates. The reflection across $y = x$ (the diagonal) involves swapping $x$ and $y$. Finally, we translate back by adding 4 to the $y$-coordinates.

4. Step 4: Apply reflection formulas for lines. The reflection of a line across another line involves finding the new slope and intercept for the reflected line. When reflecting over $y = x$, the equation of the line transforms as follows: $y = mx + b \quad \text{becomes} \quad x = my + b'$ After solving for $y$, we get the new line equation. This process needs to account for translations if the reflection line is not simply $y = x$.

5. Step 5: Calculate the reflection. Let's reflect the given line $y = 2x - 3$ across the line $y = x + 4$.

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Let me calculate the exact equation of the reflected line using the necessary transformations for reflection across $y = x + 4$.