ID: 3cdbf026
(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Algebra)
Comment: Any question with more than the two basic variables x and y gets into what I call "alphabet soup" territory. These questions present a façade of difficulty, since there can be so much to seemingly keep track of. However, they are designed to be solved using nothing more than a basic knowledge of algebra.
Method 1: Since the question asks for the value of k and one of the two points given, (0, -3), allows us to substitute 0 for x, we can directly solve for k:
a(0) + k(-3) = 6
0 - 3k = 6
k = -2
The answer must be (A).
Method 2: By the time you input the equation into Desmos and play with the sliders for a and k, you will almost assuredly have taken more time than you should have to solve the question, if you derive a solution at all.
Method 3: You could work out the slope and then solve algebraically or graph to work out the solution.
m = ((-3) - (-6))/((0) - (-2))
m = 3/2
Knowing the slope ahead of time will allow you to figure out a bit more from the original equation.
ax + ky = 6
ky = -ax + 6
y = -(a/k)x + 6/k
It must be true that -a/k = 3/2, but you could only determine that one of a or k was negative while the other was positive. Still, this information would, in combination with the given points (-2, -6) and (0, -3), be sufficient to allow you to use the slider feature on Desmos to arrive at the correct answer. First, try a = -3 and k = 2.
Nope. Notice that the y-intercept is not -3, but 3. Although not strictly necessary, you can take another 5-10 seconds to check the other combination, a = 3 and k = -2.
Perfect. Because the graph correctly models the line—i.e. corroborates known information—it must be true that k = -2, and, once again, the answer must be (A).
Finally, you could work through each combination of a and k above to see if you could derive the same solution algebraically. Pick up the work above at y = -(a/k)x + 6/k and say that a = -3 and k = 2.
y = -((-3)/(2))x + 6/(2)
Given the point (-2, -6), the harder of the two points to test, you could still evaluate.
(-6) = (3/2)(-2) + 3
-6 = -3 + 3
-6 ≠ 0
Thus, a ≠ -3 and k ≠ 2, and the only other possibility is that a = 3 and k = -2. If you are a fact-checker,
(-6) = -((3)/(-2))(-2) + 6/(-2)
-6 = -(3) + -3
-6 = -6
It might be the least efficient way of solving the problem within a 2-3 minute window, but it at least proves that k = -2 and the answer must be (A).
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