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

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

### Step 1: Determine the Slope of the Perpendicular Line
The slope of the line \(y = 5x\) is \(m_1 = 5\). 

For two lines to be perpendicular, the product of their slopes must be \(-1\). Therefore, if the slope of the given line is \(m_1 = 5\), the slope \(m_2\) of the perpendicular line must satisfy:

\[
m_1 \times m_2 = -1
\]

Substituting \(m_1 = 5\):

\[
5 \times m_2 = -1
\]

\[
m_2 = -\frac{1}{5}
\]

### Step 2: Use the Point-Slope Form to Write the Equation
The point-slope form of the equation of a line is:

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

Substituting the slope \(m = -\frac{1}{5}\) and the point \((-7, 2)\) into the equation:

\[
y - 2 = -\frac{1}{5}(x + 7)
\]

### Step 3: Simplify to Get the Slope-Intercept Form
Simplify the equation to the slope-intercept form \(y = mx + b\):

\[
y - 2 = -\frac{1}{5}x - \frac{7}{5}
\]

Add \(2\) to both sides:

\[
y = -\frac{1}{5}x + \left(-\frac{7}{5} + 2\right)
\]

\[
y = -\frac{1}{5}x + \frac{-7 + 10}{5} = -\frac{1}{5}x + \frac{3}{5}
\]

So, the equation of the line in slope-intercept form is:

\[
\boxed{y = -\frac{1}{5}x + \frac{3}{5}}
\]

Would you like more details or any other specific part of the process explained? Here are some related questions you might find helpful:

1. How do you find the equation of a line parallel to a given line?
2. What is the relationship between the slopes of parallel and perpendicular lines?
3. How do you convert the equation of a line from point-slope form to standard form?
4. How do you find the x-intercept of a line given its equation?
5. Can two lines be perpendicular if one of them has an undefined slope?
6. What is the significance of the y-intercept in the equation of a line?
7. How do you find the angle between two intersecting lines?
8. How do you determine if a point lies on a given line?

**Tip:** When finding the slope of a line perpendicular to another, simply take the negative reciprocal of the original slope.

Learn how to find the equation of a line that is perpendicular to y = 5x and passes through the point (-7, 2). This problem involves using the concept of perpendicular slopes and the point-slope form of a line equation. Suitable for high school students.

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