now write an equation that is perpendicular

To write an equation of a line that is **perpendicular** to another line, you first need to understand that the slopes of two perpendicular lines are **negative reciprocals** of each other.

Here’s a step-by-step guide:

1. Start with the equation of a line in **slope-intercept form**:  
   \[
   y = mx + b
   \]
   where \( m \) is the slope of the line.

2. If we want to write the equation of a line that is perpendicular to this, the slope of the new line will be the negative reciprocal of \( m \). This means if the slope of the original line is \( m \), the slope of the perpendicular line will be \( -\frac{1}{m} \).

### Example:

Let’s say we have a line with the equation:
\[
y = 2x + 3
\]

The slope of this line is \( m = 2 \).

For a line to be perpendicular to this one, its slope must be \( -\frac{1}{2} \).

Now, if we want to write the equation of a perpendicular line passing through a point, for example, \( (1, 4) \), we can use the point-slope form of the equation:
\[
y - y_1 = m (x - x_1)
\]
Substitute the point \( (1, 4) \) and the slope \( -\frac{1}{2} \):
\[
y - 4 = -\frac{1}{2}(x - 1)
\]
Simplifying this, we get:
\[
y - 4 = -\frac{1}{2}x + \frac{1}{2}
\]
\[
y = -\frac{1}{2}x + \frac{9}{2}
\]

Thus, the equation of the line perpendicular to \( y = 2x + 3 \) and passing through the point \( (1, 4) \) is:
\[
y = -\frac{1}{2}x + \frac{9}{2}
\]

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Would you like more details or have any questions?

Here are 5 questions you can explore further:
1. How do you find the equation of a line parallel to a given line?
2. How do you find the slope of a line given two points?
3. Can two vertical lines ever be perpendicular? Why or why not?
4. How does the slope of a horizontal line compare to that of a vertical line?
5. How can you verify if two lines are perpendicular by using their equations?

**Tip**: When converting between different forms of linear equations (like point-slope to slope-intercept), always simplify step-by-step to avoid mistakes in algebra!

Learn how to write the equation of a line that is perpendicular to another. This guide covers the slope-intercept form, point-slope form, and the concept of negative reciprocal slopes. Ideal for students in Grades 8-10.

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