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The problem asks for the equation of a line perpendicular to the line given by the equation:

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

### Step-by-step Solution:

1. **Find the slope of the given line**: The equation is in the form \(y = mx + b\), where \(m\) represents the slope. Here, the slope of the line is \(m = -\frac{2}{7}\).

2. **Find the slope of the perpendicular line**: The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. 
   
   So, the slope of the perpendicular line will be:

   \[
   m_{\text{perpendicular}} = -\left(\frac{1}{m}\right) = -\left(\frac{1}{-\frac{2}{7}}\right) = \frac{7}{2}
   \]

3. **Equation of the perpendicular line**: The line should be in slope-intercept form, \(y = mx + b\). Since we only have the slope and not the y-intercept (\(b\)), the equation of the line is:

   \[
   y = \frac{7}{2}x
   \]

This is the equation for the bike path line that is perpendicular to the given line.

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Would you like any more details or clarification? Here are five additional related questions:

1. What is the general rule for finding the slope of a perpendicular line?
2. How can you verify if two lines are perpendicular using their slopes?
3. If the bike path passes through a specific point, how would you find the exact equation of the line?
4. How do you determine the equation of a line in point-slope form?
5. Can a line be both parallel and perpendicular to another at the same time? Why or why not?

**Tip**: When finding the slope of perpendicular lines, always remember to flip the slope and change its sign.

City planners want to construct a bike path perpendicular to Seventh Street at point P. An equation for the Seventh Street line is y = -2/7 x. Find an equation for the line for the bike path.

Solve a geometry problem by finding the equation of a line perpendicular to y = -2/7x, using key algebraic concepts like slope-intercept form and perpendicular slopes. This problem is perfect for students in Grades 7-9.

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