To find the equation of a line that passes through the point $(1, -7)$ and is parallel to the line $6x + y = 8$, follow these steps:

1. Identify the slope of the given line:

The general form of a line is $Ax + By = C$. To find the slope, we convert this to the slope-intercept form $y = mx + b$, where $m$ is the slope.

Given the equation $6x + y = 8$, solve for $y$: $y = -6x + 8$ So, the slope $m$ of this line is $-6$.

2. Use the point-slope form of the equation:

The line we are looking for is parallel to this line, meaning it has the same slope. Therefore, the slope of our line is also $-6$.

The point-slope form of a line is: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is the point the line passes through, and $m$ is the slope. Using the point $(1, -7)$ and slope $-6$, we substitute into the formula: $y - (-7) = -6(x - 1)$ Simplify: $y + 7 = -6(x - 1)$ $y + 7 = -6x + 6$ $y = -6x + 6 - 7$ $y = -6x - 1$

3. Final equation:

The equation of the line is: $y = -6x - 1$

Would you like further details or have any questions?

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Here are 5 related questions to extend this topic:

1. How do you find the equation of a line perpendicular to another line?
2. What is the significance of the slope in determining parallelism between two lines?
3. How do you convert from point-slope form to slope-intercept form?
4. Can you derive the equation of a line given two points instead of one point and a slope?
5. How do you determine the y-intercept of a line given a point and the slope?

Tip: Always check the slope first when determining if two lines are parallel; if their slopes are equal, they are parallel.