write the equation of a line that is perpendicular to y = 5x and passes through (-7,2)

To write the equation of a line that is perpendicular to \( y = 5x \) and passes through the point (-7, 2), follow these steps:

### Step 1: Find the slope of the perpendicular line
- The slope of the given line \( y = 5x \) is 5.
- The slope of a line that is perpendicular to another is the negative reciprocal of the original slope. So, the slope of the perpendicular line will be:
  \[
  m = -\frac{1}{5}
  \]

### Step 2: Use the point-slope form of the equation of a line
The point-slope form of the equation of a line is:
\[
y - y_1 = m(x - x_1)
\]
Where \( m \) is the slope and \( (x_1, y_1) \) is the point the line passes through. Substituting the slope \( m = -\frac{1}{5} \) and the point \( (-7, 2) \):
\[
y - 2 = -\frac{1}{5}(x + 7)
\]

### Step 3: Simplify the equation
Now, distribute and rearrange the equation into slope-intercept form \( y = mx + b \) if needed:
\[
y - 2 = -\frac{1}{5}x - \frac{7}{5}
\]
Add 2 to both sides:
\[
y = -\frac{1}{5}x + 2 - \frac{7}{5}
\]
Convert 2 to a fraction with a denominator of 5:
\[
y = -\frac{1}{5}x + \frac{10}{5} - \frac{7}{5}
\]
Simplify:
\[
y = -\frac{1}{5}x + \frac{3}{5}
\]

So, the equation of the line that is perpendicular to \( y = 5x \) and passes through the point (-7, 2) is:
\[
y = -\frac{1}{5}x + \frac{3}{5}
\]

Would you like more details or have any other questions? Here are some related questions you might explore:

1. How do you find the equation of a line parallel to a given line through a specific point?
2. What is the general form of a linear equation, and how can you convert between forms?
3. How do you calculate the distance between two parallel lines?
4. How do perpendicular slopes relate to angles between lines?
5. Can you find the equation of a line perpendicular to a vertical line?

**Tip:** When finding perpendicular slopes, always remember the negative reciprocal relationship between the slopes.

Discover how to write the equation of a line perpendicular to y = 5x that passes through the point (-7, 2). This problem demonstrates the use of slope, the point-slope form of a line, and perpendicular slopes, suitable for students in Grades 8-10.

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