To find the equation of a line parallel to [tex]\( y = -\frac{1}{2} x - 1 \)[/tex] that passes through the point [tex]\((-4, 8)\)[/tex], we proceed as follows:
1. Identify the slope of the given line:
The given line is [tex]\( y = -\frac{1}{2} x - 1 \)[/tex]. This equation is in slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
Thus, the slope [tex]\( m \)[/tex] of the given line is [tex]\( -\frac{1}{2} \)[/tex].
2. Slope of the parallel line:
Lines that are parallel have the same slope. Therefore, the slope of the line we need to find is also [tex]\( -\frac{1}{2} \)[/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]\((x_1, y_1)\)[/tex] is a point on the line and [tex]\( m \)[/tex] is the slope.
Here, our point is [tex]\((-4, 8)\)[/tex] and our slope is [tex]\( -\frac{1}{2} \)[/tex].
Substituting the point and the slope into the point-slope form:
[tex]\[
y - 8 = -\frac{1}{2}(x + 4)
\][/tex]
4. Simplify the equation:
Let's distribute the slope [tex]\( -\frac{1}{2} \)[/tex]:
[tex]\[
y - 8 = -\frac{1}{2}x - \frac{1}{2} \cdot 4
\][/tex]
Simplify further:
[tex]\[
y - 8 = -\frac{1}{2}x - 2
\][/tex]
Now, add 8 to both sides to get the equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[
y = -\frac{1}{2}x - 2 + 8
\][/tex]
[tex]\[
y = -\frac{1}{2}x + 6
\][/tex]
5. Conclusion:
Therefore, the equation of the line that passes through [tex]\((-4, 8)\)[/tex] and is parallel to [tex]\( y = -\frac{1}{2} x - 1 \)[/tex] is:
[tex]\[
y = -\frac{1}{2} x + 6
\][/tex]
Thus, the correct choice from the given options is:
[tex]\[ y = -\frac{1}{2} x + 6 \][/tex]
