Final answer:
The equation of line k, which is perpendicular to line j (y = 5x - 8) and passes through the point (1, 7), is y = (-1/5)x + 7.
Step-by-step explanation:
To find the equation of line k that includes the point (1, 7) and is perpendicular to line j with the equation y = 5x - 8, first, we need to determine the slope of line k. Since line k is perpendicular to line j, its slope will be the negative reciprocal of the slope of line j. The slope of line j is 5, so the slope of line k is -1/5. Using the point-slope form of a line, which is y - y1 = m(x - x1), where (x1, y1) is the point on the line, and m is the slope, we plug in the point (1, 7) and the slope -1/5 to get the equation of line k.
So the equation will be y - 7 = (-1/5)(x - 1). Simplifying, we get y - 7 = (-1/5)x + 1/5. Adding 7 to both sides of the equation yields y = (-1/5)x + 7 + 1/5, which simplifies to the final equation y = (-1/5)x + 35/5. Thus, the equation of line k is y = (-1/5)x + 7.