write an equation of line L in point slope form and slope intercpet Form. L is perpendicular to y =2x and (2,4) on the graph

To find the equation of the line **L** that is perpendicular to the line \( y = 2x \) and passes through the point \( (2, 4) \), we will go step by step.

### Step 1: Find the slope of the perpendicular line
The slope of the given line \( y = 2x \) is \( m = 2 \). Since line **L** is perpendicular to this line, the slope of line **L** will be the negative reciprocal of 2. 

\[
m_{\text{L}} = -\frac{1}{2}
\]

### Step 2: Use the point-slope form
The point-slope form of a line is:

\[
y - y_1 = m(x - x_1)
\]

Where \( (x_1, y_1) \) is a point on the line (in this case \( (2, 4) \)) and \( m \) is the slope (here \( -\frac{1}{2} \)).

Substituting \( (x_1, y_1) = (2, 4) \) and \( m = -\frac{1}{2} \):

\[
y - 4 = -\frac{1}{2}(x - 2)
\]

This is the **point-slope form** of the line.

### Step 3: Convert to slope-intercept form
To convert to slope-intercept form \( y = mx + b \), expand the equation:

\[
y - 4 = -\frac{1}{2}(x - 2)
\]
\[
y - 4 = -\frac{1}{2}x + 1
\]
\[
y = -\frac{1}{2}x + 1 + 4
\]
\[
y = -\frac{1}{2}x + 5
\]

This is the **slope-intercept form** of the line.

### Final Answer:
- **Point-slope form**: \( y - 4 = -\frac{1}{2}(x - 2) \)
- **Slope-intercept form**: \( y = -\frac{1}{2}x + 5 \)

Would you like more details, or have any questions?

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Here are five related questions:
1. How do you find the negative reciprocal of a slope?
2. What does it mean for two lines to be perpendicular?
3. How do you convert point-slope form to standard form?
4. What is the equation of a line parallel to \( y = 2x \) through \( (2, 4) \)?
5. How do you calculate the distance between two points on a graph?

**Tip:** Remember, lines that are perpendicular have slopes that multiply to give -1.

write an equation of line L in point slope form and slope intercept form. L is perpendicular to y = 2x and (2,4) is on the graph.

Find the equation of a line perpendicular to y = 2x that passes through the point (2, 4). Learn how to derive the equation in point-slope form and slope-intercept form, and understand key concepts like the negative reciprocal of slopes and perpendicular lines. Suitable for Grades 8-10.

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