Sure, let's solve the problem step-by-step.
### Step 1: Identify Key Information
We are given:
- A point [tex]\((-1, -4)\)[/tex] through which the line passes.
- A line [tex]\(y = -2x\)[/tex] to which our desired line is perpendicular.
### Step 2: Determine the Slope of the Given Line
The equation of the given line is [tex]\(y = -2x\)[/tex]. The slope of this line is [tex]\(-2\)[/tex].
### Step 3: Calculate the Slope of the Perpendicular Line
The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope. Thus, the slope [tex]\(m_{\perpendicular}\)[/tex] of the line we are looking for is:
[tex]\[ m_{\perpendicular} = -\frac{1}{-2} = \frac{1}{2}. \][/tex]
### Step 4: 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]
In this case, the point [tex]\((x_1, y_1)\)[/tex] is [tex]\((-1, -4)\)[/tex] and the slope [tex]\(m\)[/tex] is [tex]\(\frac{1}{2}\)[/tex].
### Step 5: Substitute the Given Point and Slope into the Point-Slope Form
Substituting [tex]\((-1, -4)\)[/tex] and [tex]\(\frac{1}{2}\)[/tex] into the point-slope form, we get:
[tex]\[ y - (-4) = \frac{1}{2}(x - (-1)). \][/tex]
Simplifying, this becomes:
[tex]\[ y + 4 = \frac{1}{2}(x + 1). \][/tex]
### Step 6: Solve for [tex]\(y\)[/tex] to Get the Slope-Intercept Form
To get the equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], we distribute and isolate [tex]\(y\)[/tex]:
[tex]\[ y + 4 = \frac{1}{2}x + \frac{1}{2}. \][/tex]
Subtracting 4 from both sides, we obtain:
[tex]\[ y = \frac{1}{2}x + \frac{1}{2} - 4. \][/tex]
[tex]\[ y = \frac{1}{2}x - \frac{7}{2}. \][/tex]
### Final Answer
Thus, the equation of the line perpendicular to [tex]\(y = -2x\)[/tex] that passes through the point [tex]\((-1, -4)\)[/tex] is:
[tex]\[ y = \frac{1}{2}x - \frac{7}{2}. \][/tex]
The slope of this line is [tex]\( \frac{1}{2}\)[/tex] and the y-intercept is [tex]\(-\frac{7}{2}\)[/tex].
