To determine the equation of line $k$, which is parallel to line $j$ and passes through the point $(-7, -6)$, we follow these steps:

Step 1: Identify the slope of line $j$

The given equation of line $j$ is: $y - 2 = \frac{8}{9}(x + 1)$ The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope. From the given equation, the slope of line $j$ is: $m = \frac{8}{9}$

Since line $k$ is parallel to line $j$, its slope is also $\frac{8}{9}$.

Step 2: Use the point-slope form for line $k$

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ Substitute $m = \frac{8}{9}$, $x_1 = -7$, and $y_1 = -6$: $y - (-6) = \frac{8}{9}(x - (-7))$ Simplify: $y + 6 = \frac{8}{9}(x + 7)$

Step 3: Convert to slope-intercept form

Distribute $\frac{8}{9}$ on the right-hand side: $y + 6 = \frac{8}{9}x + \frac{8}{9}(7)$ $y + 6 = \frac{8}{9}x + \frac{56}{9}$ Subtract $6$ (or $\frac{54}{9}$) from both sides: $y = \frac{8}{9}x + \frac{56}{9} - \frac{54}{9}$ $y = \frac{8}{9}x + \frac{2}{9}$

Final Answer:

The equation of line $k$ is: $y = \frac{8}{9}x + \frac{2}{9}$

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Would you like further clarification or details about these steps? Here are five related questions:

1. How can we verify if two lines are parallel or perpendicular using their slopes?
2. What would the equation of a line perpendicular to line $j$ look like?
3. How do we determine the equation of a line in point-slope form?
4. How does the slope-intercept form differ from the standard form of a line?
5. What are some applications of parallel lines in geometry and real-world contexts?

Tip: Always check your slope and points carefully when converting between forms to avoid calculation errors.