Find the equation of the line that is parallel to $$ y=\frac{1}{2} x-4 $$ and passes through the point $$ (1,-5) $$.

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To find the equation of a line that is parallel to the line given by $$ y = \frac{1}{2} x - 4 $$ and passes through the point $$ (1, -5) $$, we first note that parallel lines have the same slope.

The slope of the given line $$ y = \frac{1}{2} x - 4 $$ is $$ \frac{1}{2} $$. Therefore, the slope of the line we want to find is also $$ \frac{1}{2} $$.

We can use the point-slope form of the equation of a line, which is given by:

$$
y - y_1 = m(x - x_1)
$$

where $$ m $$ is the slope, and $$ (x_1, y_1) $$ is a point on the line. Here, $$ m = \frac{1}{2} $$ and the point is $$ (1, -5) $$.

Substituting these values into the point-slope form:

$$
y - (-5) = \frac{1}{2}(x - 1)
$$

This simplifies to:

$$
y + 5 = \frac{1}{2}(x - 1)
$$

Next, we distribute the $$ \frac{1}{2} $$:

$$
y + 5 = \frac{1}{2}x - \frac{1}{2}
$$

Now, we isolate $$ y $$ by subtracting 5 from both sides:

$$
y = \frac{1}{2}x - \frac{1}{2} - 5
$$

To combine the constants, we convert -5 to a fraction with a denominator of 2:

$$
-5 = -\frac{10}{2}
$$

Thus, we have:

$$
y = \frac{1}{2}x - \frac{1}{2} - \frac{10}{2}
$$

This simplifies to:

$$
y = \frac{1}{2}x - \frac{11}{2}
$$

Therefore, the equation of the line that is parallel to $$ y = \frac{1}{2} x - 4 $$ and passes through the point $$ (1, -5) $$ is:

$$
\boxed{y = \frac{1}{2}x - \frac{11}{2}}
$$
The equation of the line is $$ y = \frac{1}{2}x - \frac{11}{2} $$.

Find the equation of the line that is parallel to \( y=\frac{1}{2} x-4 \) and passes through the point \( (1,-5) \).

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