Question 592·Medium·Algebra

In $y=mx+b$, the slope $m$ tells you how much $y$ changes when $x$ increases by $1$. The given line is $y=\tfrac{1}{2}x+7$, so its slope is the number multiplying $x$, $\tfrac{1}{2}$. The $7$ sets where the line crosses the $y$-axis, not its slope.

Line $\ell$ is perpendicular, meaning it meets the given line at a right angle. Its slope must be the negative reciprocal: $\tfrac{1}{2}$ means right $2$, up $1$; after a right-angle turn, that becomes right $1$, down $2$. So line $\ell$ needs slope $-2$. For perpendicular slopes, flip the fraction and change the sign.

Slope is the change in $y$ divided by the change in $x$. Take the second point minus the first in both places: the $y$ change is $k-4$, and the $x$ change is $5-(-3)$. Because this is line $\ell$, its slope must equal $-2$:

Type $\frac{k-4}{5-(-3)}\sim-2$ in Desmos. The $\sim$ asks Desmos to find the value of $k$ that makes the two slopes equal. Under PARAMETERS, it reports $k=-12$. That is the missing $y$-coordinate in $(5,k)$. Choice B.