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------------------------------------------------ Find the equation of the line parallel to [tex]$y=\frac{1}{2} x-4$[/tex] that passes through the point [tex]$(3,-2)$[/tex] in slope-intercept form.

A. [tex]y=4 x+\frac{1}{2}[/tex]
B. [tex]y=\frac{1}{2} x-\frac{7}{2}[/tex]
C. [tex]y=4 x-\frac{1}{2}[/tex]

To find the equation of a line that is parallel to [tex]$ y = \frac{1}{2}x - 4 $[/tex] and passes through the point [tex]$ (3, -2) $[/tex], you'll follow these steps:

1. Identify the Slope:
The equation given is in the slope-intercept form [tex]$ y = mx + b $[/tex], where [tex]$ m $[/tex] is the slope. Here, [tex]$ m = \frac{1}{2} $[/tex].

2. Understand Parallel Lines:
Lines that are parallel have the same slope. Therefore, the slope of our new line will also be [tex]$ \frac{1}{2} $[/tex].

3. Use the Point-Slope Form:
Since the new line passes through the point [tex]$ (3, -2) $[/tex], we use the point-slope form of a line, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Plug in the point [tex]$ (3, -2) $[/tex] and the slope [tex]$ \frac{1}{2} $[/tex]:
[tex]\[
y - (-2) = \frac{1}{2}(x - 3)
\][/tex]
Simplify this to:
[tex]\[
y + 2 = \frac{1}{2}x - \frac{3}{2}
\][/tex]

4. Convert to Slope-Intercept Form:
Isolate [tex]$ y $[/tex] to express the equation in slope-intercept form [tex]$ y = mx + b $[/tex]:
[tex]\[
y = \frac{1}{2}x - \frac{3}{2} - 2
\][/tex]
Calculate the value:
[tex]\[
y = \frac{1}{2}x - \frac{3}{2} - \frac{4}{2}
\][/tex]
[tex]\[
y = \frac{1}{2}x - \frac{7}{2}
\][/tex]

5. Final Equation:
The equation of the line parallel to [tex]$ y = \frac{1}{2}x - 4 $[/tex] and passing through [tex]$ (3, -2) $[/tex] is:
[tex]\[
y = \frac{1}{2}x - \frac{7}{2}
\][/tex]

The equation of the line that satisfies the given conditions is [tex]$ y = \frac{1}{2}x - \frac{7}{2} $[/tex].