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The problem asks to find the equation of a new path that is perpendicular to the given path, represented by the equation \( y = -2x - 7 \), and intersects at the point \((-2, -3)\).

### Step-by-Step Solution:

1. **Find the slope of the original line:**
   The equation of the original path is in the slope-intercept form, \( y = mx + b \), where \( m \) is the slope. Here, the slope of the original path is:
   \[
   m_1 = -2
   \]

2. **Find the slope of the perpendicular line:**
   Two lines are perpendicular if the product of their slopes equals \(-1\). If \( m_1 \) is the slope of the original line, the slope of the perpendicular line, \( m_2 \), can be found using the relation:
   \[
   m_1 \cdot m_2 = -1
   \]
   Substituting \( m_1 = -2 \):
   \[
   -2 \cdot m_2 = -1 \quad \Rightarrow \quad m_2 = \frac{1}{2}
   \]

3. **Use point-slope form to write the equation:**
   The perpendicular line passes through the point \((-2, -3)\). The point-slope form of a line is:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( m = \frac{1}{2} \), \( x_1 = -2 \), and \( y_1 = -3 \):
   \[
   y - (-3) = \frac{1}{2}(x - (-2))
   \]
   Simplifying:
   \[
   y + 3 = \frac{1}{2}(x + 2)
   \]
   \[
   y + 3 = \frac{1}{2}x + 1
   \]
   \[
   y = \frac{1}{2}x + 1 - 3
   \]
   \[
   y = \frac{1}{2}x - 2
   \]

Thus, the equation of the new path is:
\[
y = \frac{1}{2}x - 2
\]

### Graphical Check:
The graph of the original line with a slope of \(-2\) is visible in the image, and we derived the equation of the new line, which has a slope of \(\frac{1}{2}\). The two lines are perpendicular as required.

Would you like further details, or do you have any questions about the process?

Here are five related questions to expand your understanding:
1. How would you find the equation if the paths were parallel instead of perpendicular?
2. Can you verify that two lines are perpendicular by looking at their angles?
3. What would the equation of the line be if the point of intersection were different, say \((1, 2)\)?
4. How does the slope of a line affect its steepness and direction on a graph?
5. What changes in the equation if the y-intercept of the original path were different?

Tip: Always remember, for perpendicular lines, the slopes are negative reciprocals of each other!

A walking path across a park is represented by the equation y = -2x - 7. A new path will be built perpendicular to this path. The paths will intersect at the point (-2, -3). Identify the equation that represents the new path.

Learn how to find the equation of a line perpendicular to y = -2x - 7 that passes through the point (-2, -3). This problem involves understanding linear equations, the concept of slopes, and using the point-slope form.

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