Certainly! To find the equation of a line that is parallel to a given line and passes through a specific point, we can follow these steps:
1. Identify the slope of the given line.
The given line is [tex]\( y = \frac{1}{2}x - 4 \)[/tex]. The slope of this line is [tex]\(\frac{1}{2}\)[/tex].
2. Determine the slope of the parallel line.
Lines that are parallel have the same slope, so the slope of the new line will also be [tex]\(\frac{1}{2}\)[/tex].
3. Use the point-slope form of the equation of a line.
The point-slope form of a line's 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. We know the slope [tex]\( m = \frac{1}{2} \)[/tex] and the line passes through the point [tex]\( (4, 5) \)[/tex].
4. Substitute the known values into the point-slope form.
Plugging in the slope and the coordinates of the point [tex]\( (4, 5) \)[/tex]:
[tex]\[
y - 5 = \frac{1}{2}(x - 4)
\][/tex]
5. Simplify the equation to slope-intercept form [tex]\( y = mx + b \)[/tex].
To do this, we need to distribute [tex]\(\frac{1}{2}\)[/tex] and then solve for [tex]\( y \)[/tex]:
[tex]\[
y - 5 = \frac{1}{2}x - 2
\][/tex]
Now, add 5 to both sides to isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{2}x - 2 + 5
\][/tex]
Combine like terms:
[tex]\[
y = \frac{1}{2}x + 3
\][/tex]
So, the equation of the line that is parallel to [tex]\(y = \frac{1}{2}x - 4\)[/tex] and passes through the point [tex]\( (4, 5) \)[/tex] is:
[tex]\[
y = \frac{1}{2}x + 3
\][/tex]
Therefore, the final equation you are looking for is:
[tex]\[
y = \frac{1}{2} x + 3
\][/tex]
