A line y=2x−3*y*=2*x*−3 is reflected across the line y=x+4*y*=*x*+4. Determine the equation of the reflection of the line.

**What is required?**

Find the reflection of the line across another line.

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 \( y = x + 4 \).

A line y = 2x - 3 is reflected across the line y = x + 4. Determine the equation of the reflection of the line.

Learn how to reflect the line y = 2x - 3 across another line y = x + 4. This problem involves key concepts from coordinate geometry, including line reflections and transformation properties. Suitable for students in Grades 10-12.

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