We are tasked with finding the equation of line $k$, which is parallel to line $j$. The equation for line $j$ is given as:

$y = \frac{1}{2}x - 4$

Step 1: Identify the slope of line $j$

The slope-intercept form of a line is:

$y = mx + b$

where $m$ is the slope. From the equation $y = \frac{1}{2}x - 4$, we see that the slope $m$ is $\frac{1}{2}$.

Step 2: Use the same slope for line $k$

Since line $k$ is parallel to line $j$, they share the same slope. Therefore, the slope of line $k$ is also $\frac{1}{2}$.

Step 3: Use the point-slope form

We are given that line $k$ passes through the point $(-8, -5)$. To find the equation of line $k$, we use the point-slope form of a line:

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

where $m$ is the slope and $(x_1, y_1)$ is the point the line passes through. Substituting the values:

• $m = \frac{1}{2}$
• $x_1 = -8$
• $y_1 = -5$

we get:

$y - (-5) = \frac{1}{2}(x - (-8))$

This simplifies to:

$y + 5 = \frac{1}{2}(x + 8)$

Step 4: Solve for $y$

Now we solve for $y$ to put the equation into slope-intercept form:

$y + 5 = \frac{1}{2}x + \frac{1}{2}(8)$ $y + 5 = \frac{1}{2}x + 4$ $y = \frac{1}{2}x + 4 - 5$ $y = \frac{1}{2}x - 1$

Final Answer:

The equation of line $k$ is:

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

Would you like more details on any of the steps?

Here are some additional questions related to this problem:

1. How do you find the slope of a line given two points on the line?
2. What is the difference between parallel and perpendicular lines in terms of their slopes?
3. How can you convert the equation of a line from point-slope form to slope-intercept form?
4. What would the equation of line $k$ be if it were perpendicular to line $j$ instead?
5. How do you check if a given point lies on a line?

Tip: When two lines are parallel, their slopes are always equal!