Line k is defined by y = 3 x + 15 . Line j is perpendicular

Choice A is correct. It's given that line $j$ is perpendicular to line $k$ in the xy-plane. It follows that the slope of line $j$ is the opposite reciprocal of the slope of line $k$. The equation for line $k$ is written in slope-intercept form $y=mx+b$, where $m$ is the slope of the line and $b$ is the y-coordinate of the y-intercept of the line. It follows that the slope of line $k$ is $3$. The opposite reciprocal of a number is $-1$ divided by the number. Thus, the opposite reciprocal of $3$ is $-\frac{1}{3}$. Therefore, the slope of line $j$ is $-\frac{1}{3}$.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.