To determine the equation of a line that is parallel to a given line and passes through a specific point, we need to follow these steps:
Step 1: Identify the slope of the given line.
The given line's equation is [tex]\( y = \frac{1}{2}x + 6 \)[/tex].
This equation is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. From the equation, we see that the slope [tex]\( m \)[/tex] is [tex]\( \frac{1}{2} \)[/tex].
Step 2: Find the slope of the parallel line.
Lines that are parallel have the same slope. Therefore, the slope of the line parallel to [tex]\( y = \frac{1}{2} x + 6 \)[/tex] is also [tex]\( \frac{1}{2} \)[/tex].
Step 3: Use the point through which the parallel line passes.
We are given the point [tex]\( (0, -2) \)[/tex]. We need to use this point to find the y-intercept [tex]\( b \)[/tex] of the new line.
Step 4: Substitute the slope and the coordinates of the point into the slope-intercept equation [tex]\( y = mx + b \)[/tex].
To find the y-intercept [tex]\( b \)[/tex], substitute the slope ([tex]\( \frac{1}{2} \)[/tex]) and the coordinates of the point [tex]\( (0, -2) \)[/tex] into the equation [tex]\( y = mx + b \)[/tex]:
[tex]\[
-2 = \left(\frac{1}{2} \times 0\right) + b
\][/tex]
Step 5: Solve for [tex]\( b \)[/tex].
[tex]\[
-2 = 0 + b \implies b = -2
\][/tex]
Step 6: Write the equation of the parallel line.
Now that we have [tex]\( m = \frac{1}{2} \)[/tex] and [tex]\( b = -2 \)[/tex], the equation of the line in slope-intercept form is:
[tex]\[
y = \frac{1}{2} x - 2
\][/tex]
Conclusion:
Among the given choices, the equation [tex]\( y = \frac{1}{2} x - 2 \)[/tex] is the correct one.
Thus, the equation of the line parallel to [tex]\( y = \frac{1}{2} x + 6 \)[/tex] and passing through the point [tex]\( (0, -2) \)[/tex] is:
[tex]\[
y = \frac{1}{2} x - 2
\][/tex]
So the correct choice is:
[tex]\[
\boxed{y = \frac{1}{2} x - 2}
\][/tex]
