Find the equation of the line that is perpendicular to the line y=-2x-7 and passes through the origin

To find the equation of the line that is perpendicular to the line \( y = -2x - 7 \) and passes through the origin, we'll follow these steps:

### 1. Identify the slope of the given line.
The given line is \( y = -2x - 7 \). The slope-intercept form of a line is \( y = mx + b \), where \( m \) is the slope. For this line, the slope \( m_1 = -2 \).

### 2. Determine the slope of the perpendicular line.
The slope of a line that is perpendicular to another line is the negative reciprocal of the slope of the original line. If the slope of the original line is \( m_1 \), then the slope of the perpendicular line \( m_2 \) is given by:

\[
m_2 = -\frac{1}{m_1}
\]

For the given line, the slope \( m_1 = -2 \). So, the slope of the perpendicular line \( m_2 \) is:

\[
m_2 = -\frac{1}{-2} = \frac{1}{2}
\]

### 3. Use the slope-point form to write the equation of the perpendicular line.
The perpendicular line passes through the origin, which has coordinates \( (0, 0) \). The slope-point form of the equation of a line is:

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

Substitute \( m = \frac{1}{2} \), \( x_1 = 0 \), and \( y_1 = 0 \):

\[
y - 0 = \frac{1}{2}(x - 0)
\]

Simplify this equation:

\[
y = \frac{1}{2}x
\]

### Final Answer:
The equation of the line that is perpendicular to \( y = -2x - 7 \) and passes through the origin is:

\[
y = \frac{1}{2}x
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the general formula for finding the equation of a line?
2. How do you find the equation of a line parallel to a given line?
3. What is the significance of the slope in the context of perpendicular lines?
4. How do you determine the point of intersection between two lines?
5. How would the equation change if the line passed through a different point instead of the origin?
6. Can you graph the line \( y = \frac{1}{2}x \) and the line \( y = -2x - 7 \)?
7. What is the relationship between the slopes of parallel lines?
8. How does the concept of perpendicularity apply to vectors?

**Tip:** When working with perpendicular lines, always remember that their slopes are negative reciprocals of each other!

Learn how to find the equation of a line perpendicular to y = -2x - 7 that passes through the origin. This problem involves using the concepts of slopes, perpendicular lines, and the slope-point form of a line equation. Suitable for high school students.

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