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Pencerminan garis y = 5x - 1 terhadap garis x = 3 menghasilkan garis

The problem in the image is asking about the reflection of the line \( y = 5x - 1 \) over the line \( x = 3 \), resulting in a new line equation.

### Step-by-step solution:

1. **Identify the line of reflection**: The line of reflection is \( x = 3 \), which is a vertical line. When reflecting a line over a vertical line, the slope of the original line is preserved, but the position (y-intercept or shift) may change.

2. **Find the reflected points**: To find the equation of the reflected line, we need to reflect any two points on the line \( y = 5x - 1 \) over the line \( x = 3 \).

   - Choose two points on \( y = 5x - 1 \). For instance:
     - When \( x = 0 \), \( y = 5(0) - 1 = -1 \), so one point is \( (0, -1) \).
     - When \( x = 1 \), \( y = 5(1) - 1 = 4 \), so another point is \( (1, 4) \).

3. **Reflect the points over \( x = 3 \)**:
   - To reflect a point \( (x_1, y_1) \) over a vertical line \( x = 3 \), the x-coordinate changes according to the formula:
     \[
     x' = 2 \cdot 3 - x_1 = 6 - x_1
     \]
     The y-coordinate stays the same.

   - Reflect \( (0, -1) \) over \( x = 3 \):
     \[
     x' = 6 - 0 = 6 \quad \text{so the reflected point is} \quad (6, -1)
     \]

   - Reflect \( (1, 4) \) over \( x = 3 \):
     \[
     x' = 6 - 1 = 5 \quad \text{so the reflected point is} \quad (5, 4)
     \]

4. **Find the equation of the reflected line**:
   - Now we have two points on the reflected line: \( (6, -1) \) and \( (5, 4) \).
   - The slope \( m \) of the reflected line is the same as the original line, which is 5.
   - Using the point-slope form of the equation of a line:
     \[
     y - y_1 = m(x - x_1)
     \]
     Substituting \( (6, -1) \) and \( m = 5 \):
     \[
     y + 1 = 5(x - 6)
     \]
     \[
     y + 1 = 5x - 30
     \]
     \[
     y = 5x - 31
     \]

### Conclusion:
The equation of the line after reflecting \( y = 5x - 1 \) over \( x = 3 \) is \( y = 5x - 31 \).

Would you like any further details on the steps? Feel free to ask!

Here are 5 related questions to deepen your understanding:
1. How do you reflect a point over a vertical or horizontal line in general?
2. What would happen if you reflect a line over a horizontal line instead of a vertical one?
3. How can you derive the slope-intercept form from two given points?
4. How does reflecting a point affect its coordinates geometrically?
5. How do we verify the correctness of the reflection in this case?

**Tip**: When reflecting over vertical lines, always remember that the slope remains unchanged while only the intercept shifts based on the reflected points.

Learn how to find the equation of a line reflected over another line. This problem involves reflecting the line y = 5x - 1 over the vertical line x = 3, using the reflection formula and slope-intercept concepts. Suitable for students in Grades 9-11.

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