Question

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.

Answer 1

To find the equation of the new path that is perpendicular to the given path represented by the equation \( y = -2x - 7 \), we first need to determine the slope of the new path.

1. **Identify the slope of the original path**:
The equation \( y = -2x - 7 \) is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope. Here, the slope \( m \) is -2.

2. **Find the slope of the perpendicular path**:
The slopes of two perpendicular lines are negative reciprocals of each other. Therefore, the slope of the new path will be:
\[
m_{\text{new}} = -\frac{1}{m_{\text{original}}} = -\frac{1}{-2} = \frac{1}{2}
\]

3. **Use the point of intersection**:
We know that the new path intersects the original path at the point (-2, -3). We can use this point and the slope we just calculated to write the equation of the new path using the point-slope form of a line, which is given by:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the point (-2, -3) and the slope \( \frac{1}{2} \):
\[
y - (-3) = \frac{1}{2}(x - (-2))
\]
Simplifying this gives:
\[
y + 3 = \frac{1}{2}(x + 2)
\]

4. **Distribute and rearrange to find the equation in slope-intercept form**:
\[
y + 3 = \frac{1}{2}x + 1
\]
Subtracting 3 from both sides:
\[
y = \frac{1}{2}x + 1 - 3
\]
Thus, the equation becomes:
\[
y = \frac{1}{2}x - 2
\]

So the equation that represents the new path is:
\[
\boxed{y = \frac{1}{2}x - 2}
\]