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Line L in the figure below is parallel to the line y=2x+2. Find the coordinates of the point P.


The diagram shows two parallel lines: the given line \( y = 2x + 2 \) and line \( L \), which passes through point \( P \). There is also a point \( (0, -2) \) on line \( L \), which we can use to find the equation of line \( L \) since the lines are parallel and have the same slope.

### Steps to find the coordinates of \( P \):

1. **Identify the slope of line \( L \)**: 
   Since line \( L \) is parallel to \( y = 2x + 2 \), both lines share the same slope. Therefore, the slope of line \( L \) is also 2.

2. **Write the equation of line \( L \)**:
   Using the point-slope form of a line \( y - y_1 = m(x - x_1) \), where \( m = 2 \) (the slope) and the point \( (0, -2) \) lies on line \( L \):
   \[
   y - (-2) = 2(x - 0)
   \]
   Simplifying:
   \[
   y + 2 = 2x
   \]
   \[
   y = 2x - 2
   \]
   So, the equation of line \( L \) is \( y = 2x - 2 \).

3. **Determine the coordinates of point \( P \)**:
   Point \( P \) lies on both the line \( L \) and the x-axis. Since \( P \) is on the x-axis, its \( y \)-coordinate is 0. We can use this information to find the \( x \)-coordinate by substituting \( y = 0 \) into the equation of line \( L \):
   \[
   0 = 2x - 2
   \]
   Solving for \( x \):
   \[
   2x = 2 \quad \Rightarrow \quad x = 1
   \]
   Therefore, the coordinates of point \( P \) are \( (1, 0) \).

### Final Answer:
The coordinates of point \( P \) are \( \boxed{(1, 0)} \).

Let me know if you'd like any more details or explanations!

Here are 5 follow-up questions related to this problem:
1. How do you calculate the slope of a line given two points?
2. What is the relationship between the slopes of parallel lines?
3. How do you find the equation of a line given its slope and a point on the line?
4. What happens to the slope when two lines are perpendicular?
5. How do you determine the distance between two parallel lines?

**Tip**: Parallel lines have identical slopes, which means they will never intersect unless they are the same line.

Line L in the figure below is parallel to the line y=2x+2. Find the coordinates of the point P.

Explore how to find the coordinates of point P on a line parallel to y=2x+2. Using the concepts of parallel lines and the point-slope form, this problem provides a step-by-step solution. Suitable for students in Grades 8-10.

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