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.