Test 3 Section 4 #26

Fun fact and major shortcut: when a line passes through the origin, the coordinates of each point are in the same ratio. For example, if you knew a line passed through (3, 6) and (30, 60), you’d know that line passed through the origin. That’s because when a line passes through the origin, its equation will simply be ymx + 0, or simply ymx. Put another way, y will be directly proportional to x.

We can use that fact to solve quickly:

    \begin{align*}\frac{2}{k}&=\frac{k}{32}\\\\k^2&=64\\k&=\pm 8\end{align*}

Of course, because there are no negative answer choices, we choose C for 8.

The other way to go here would be to backsolve by just trying to draw the points. You can cross off A right away because any line that goes through (2, 0) and (0, 32) can’t also go through the origin! Even a rough drawing should convince you that the line containing(2, 16) and (16, 32) doesn’t go through the origin. Things get a little trickier with (2, 4) and (4, 32) because it might look CLOSE, but hopefully the right answer looks even closer.