To determine the equation of a line perpendicular to the line [tex]\(WX\)[/tex] and passing through the point [tex]\((-1, -2)\)[/tex], we need to follow several steps:
1. Convert the given line equation to slope-intercept form:
The equation of line [tex]\(WX\)[/tex] is given by:
[tex]\[
2x + y = -5
\][/tex]
To express this in slope-intercept form [tex]\(y = mx + b\)[/tex], we need to solve for [tex]\(y\)[/tex]:
[tex]\[
y = -2x - 5
\][/tex]
Here, the slope [tex]\(m\)[/tex] of line [tex]\(WX\)[/tex] is [tex]\(-2\)[/tex].
2. Find the slope of the perpendicular line:
The slope of a line perpendicular to another is the negative reciprocal of the original line's slope. Therefore, the slope [tex]\(m_{\perpendicular}\)[/tex] of the perpendicular line is:
[tex]\[
m_{\perpendicular} = -\frac{1}{-2} = \frac{1}{2}
\][/tex]
3. Use the point-slope form to find the equation of the perpendicular line:
The point-slope form of a line's equation is given by:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Given the point [tex]\((-1, -2)\)[/tex], we have:
[tex]\[
x_1 = -1 \quad \text{and} \quad y_1 = -2
\][/tex]
Substituting the point [tex]\((-1, -2)\)[/tex] and the slope [tex]\(\frac{1}{2}\)[/tex] into the point-slope form equation, we get:
[tex]\[
y - (-2) = \frac{1}{2}(x - (-1))
\][/tex]
Simplifying this, we get:
[tex]\[
y + 2 = \frac{1}{2}(x + 1)
\][/tex]
4. Simplify to slope-intercept form:
To express the equation in slope-intercept form [tex]\(y = mx + b\)[/tex], distribute the slope:
[tex]\[
y + 2 = \frac{1}{2}x + \frac{1}{2}
\][/tex]
Then, isolate [tex]\(y\)[/tex] by subtracting 2 from both sides:
[tex]\[
y = \frac{1}{2}x + \frac{1}{2} - 2
\][/tex]
[tex]\[
y = \frac{1}{2}x - \frac{3}{2}
\][/tex]
Thus, the equation of the line perpendicular to line [tex]\(WX\)[/tex] and passing through the point [tex]\((-1, -2)\)[/tex] is:
[tex]\[
y = 0.5x - 1.5
\][/tex]
