To find the equation of the new path that is perpendicular to the original path, we can follow these steps:
1. Identify the slope of the original path:
The equation of the original path is given by [tex]\( y = -2x - 7 \)[/tex]. The slope of this line, often denoted as [tex]\( m \)[/tex], is [tex]\(-2\)[/tex].
2. Determine the slope of the perpendicular path:
If two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex]. Therefore, the slope of the new path, which we'll call [tex]\( m' \)[/tex], is the negative reciprocal of the original slope.
[tex]\[
m' = -\frac{1}{m} = -\frac{1}{-2} = 0.5
\][/tex]
3. Use the point-slope form of the line equation:
The point-slope form of a line equation is given by [tex]\( y - y_1 = m'(x - x_1) \)[/tex], where [tex]\( m' \)[/tex] is the slope and [tex]\((x_1, y_1)\)[/tex] is the point of intersection.
Given that our point of intersection is [tex]\((-2, -3)\)[/tex], we substitute [tex]\(m' = 0.5\)[/tex], [tex]\(x_1 = -2\)[/tex], and [tex]\(y_1 = -3\)[/tex] into the point-slope form:
[tex]\[
y - (-3) = 0.5(x - (-2))
\][/tex]
Simplify the expression:
[tex]\[
y + 3 = 0.5(x + 2)
\][/tex]
4. Convert to slope-intercept form:
To convert the equation to slope-intercept form ([tex]\( y = mx + b \)[/tex]), we distribute [tex]\(0.5\)[/tex] on the right side:
[tex]\[
y + 3 = 0.5x + 1
\][/tex]
Subtract 3 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 0.5x + 1 - 3
\][/tex]
[tex]\[
y = 0.5x - 2
\][/tex]
Therefore, the equation of the new path that is perpendicular to the original path and intersects it at the point [tex]\((-2, -3)\)[/tex] is:
[tex]\[
y = 0.5x - 2
\][/tex]
