Write an equation for line L (2,4) in​ point-slope form and​ slope-intercept form.
L is perpendicular to
y equals 2 x

To solve this problem, we need to find the equation of line L, which passes through the point \((2, 4)\) and is perpendicular to the line \(y = 2x\).

### Step 1: Determine the slope of the line perpendicular to \(y = 2x\)

The equation \(y = 2x\) is in slope-intercept form, \(y = mx + b\), where \(m\) is the slope. The slope of the given line is \(m = 2\).

For two lines to be perpendicular, the product of their slopes must equal \(-1\). If the slope of the first line is \(m_1\), and the slope of the second line is \(m_2\), then:
\[
m_1 \cdot m_2 = -1
\]
Since the slope of the given line is \(2\), we can find the slope of the line perpendicular to it by solving for \(m_2\):
\[
2 \cdot m_2 = -1 \quad \Rightarrow \quad m_2 = -\frac{1}{2}
\]
So, the slope of the perpendicular line (line L) is \(-\frac{1}{2}\).

### Step 2: Write the equation in point-slope form

The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. Since the line passes through the point \((2, 4)\) and has a slope of \(-\frac{1}{2}\), we substitute these values into the point-slope form:
\[
y - 4 = -\frac{1}{2}(x - 2)
\]
This is the equation of the line in point-slope form.

### Step 3: Convert to slope-intercept form

To convert to slope-intercept form \(y = mx + b\), we simplify the point-slope equation:
\[
y - 4 = -\frac{1}{2}(x - 2)
\]
Distribute the \(-\frac{1}{2}\):
\[
y - 4 = -\frac{1}{2}x + 1
\]
Add 4 to both sides to solve for \(y\):
\[
y = -\frac{1}{2}x + 5
\]
Thus, the equation of the line in slope-intercept form is:
\[
y = -\frac{1}{2}x + 5
\]

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

Would you like any further details or have any questions?

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Here are 5 related questions for further exploration:

1. How do you find the equation of a line given two points?
2. What is the relationship between parallel and perpendicular lines?
3. How do you convert a point-slope form equation to standard form?
4. How can you determine if two lines are perpendicular from their slopes?
5. What is the meaning of the y-intercept in the slope-intercept form?

**Tip:** To quickly check if two lines are perpendicular, multiply their slopes. If the result is \(-1\), they are perpendicular.

Write an equation for line L (2,4) in point-slope form and slope-intercept form. L is perpendicular to y = 2x.

Learn how to find the equation of a line passing through the point (2,4) that is perpendicular to y = 2x. This solution provides detailed steps for finding both point-slope and slope-intercept forms, covering key algebraic concepts such as perpendicular slopes and linear equations. Suitable for students in Grades 8-10.

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