Math: 0b46bad5

ID: 0b46bad5

(SAT Suite Question Bank > Find Questions > Assessment: SAT + Test: Math + Domain: Algebra)

Comment: This question is more logic-based (conceptual) than skill-based and does not require solving.

Method 1: There are only two vital pieces of information in this problem: the equation, of course, and the inequality 0 < a < b, which tells us that the two unknowns must be positive. (We do not know whether they are integers, but they cannot be either 0 or negative.) Think about the equation:

ax + by = b

In y = mx + b terms, this would mean the slope, m, would have to be negative and b/b would equal 1. We can look at the graphs and rule out (B) for having the wrong y-intercept. Now, if, as the inequality tells us, a < b, then a little (perhaps mental) manipulation will reveal a certain truth:

ax + by = b

by = -ax + b

y = -(a/b)x + 1

Because the slope represents "rise over run," it must be true that the graph will "rise" less than it "runs." In answer choice (A), the two are equal, as though a = b, and that cannot be right. (C) looks good, while (D) reverses the proper relationship and shows a steeply declining line (i.e. a > b). Thus, the correct answer must be (C).

Method 2: Write the equation exactly as is, adding sliders for a and b, and make sure to observe the given inequality.

You can easily recreate answer choice (C) by leaving the slider for a alone and sliding b to 2. To be honest, this might be faster than method 1 above and can ensure that you do not mix anything up in your head.

Method 3: You can always select values for unknowns in an open-ended question (one that follows the frame could be true), as long as you adhere to any given constraints. Here, the two unknowns must be positive, and a < b. How about 3 and 10, respectively, just to prove a point?

ax + by = b

(3)x + (10)y = (10)

10y = -3x + 10

y = -(3/10)x + 1

The only graph that depicts a slope like the one above—ignoring the actual numbers on the graph—is (C), making it the correct answer.