To determine the equation of a line parallel to the given line [tex]\( y = \frac{1}{4} x - 2 \)[/tex] and passing through the point [tex]\( (4, -2) \)[/tex], follow these steps:
1. Identify the slope of the given line:
- The equation of the line is [tex]\( y = \frac{1}{4} x - 2 \)[/tex].
- The slope-intercept form of a line's equation is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope.
- Therefore, the slope [tex]\( m \)[/tex] of this line is [tex]\( \frac{1}{4} \)[/tex].
2. Use the slope of the parallel line:
- Parallel lines have the same slope. So the slope of the required parallel line is also [tex]\( \frac{1}{4} \)[/tex].
3. Use the point-slope form to find the equation:
- The point-slope form of a line's equation is [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.
- Given point [tex]\( (4, -2) \)[/tex] and slope [tex]\( \frac{1}{4} \)[/tex], substitute these values into the point-slope form:
[tex]\[
y - (-2) = \frac{1}{4} (x - 4)
\][/tex]
4. Simplify the equation:
- Start by simplifying the left side:
[tex]\[
y + 2 = \frac{1}{4}(x - 4)
\][/tex]
- Distribute the slope [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[
y + 2 = \frac{1}{4} x - 1
\][/tex]
- Subtract 2 from both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{4} x - 1 - 2
\][/tex]
- Simplify further:
[tex]\[
y = \frac{1}{4} x - 3
\][/tex]
Therefore, the equation of the line parallel to the given line and passing through the point [tex]\( (4, -2) \)[/tex] is [tex]\( y = \frac{1}{4} x - 3 \)[/tex].
The correct answer is:
[tex]\[ \boxed{y = \frac{1}{4} x - 3} \][/tex]
