Math: 3f8a701b

ID: 3f8a701b

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

Comment: Watch those assumptions.

Method 1: At a glance, this appears to be a five-second question: no solutions to an equation with algebra on both sides means that the algebraic part will have to cancel out and the numeric part will have to be different. But I have seen plenty of 700-level students answer the question quickly and miss it because they overlook one crucial part of the equation.

9x + 5 = a(x + b)

Now, obviously to yield no solutions, a will have to equal 9, so statement I must be true, and we can eliminate answer choice (A). But remember that once distributed, that 9 will turn into 9b, so we cannot answer the question with b in mind only. Distribute the 9 and work out the arithmetic, knowing that the two sides cannot be equal.

9x + 5 ≠ (9)(x + b)

9x + 5 ≠ 9x + 9b

5 ≠ 9b

5/9 ≠ b

Thus, statement III must also be true, and the correct answer is (D).

Method 2: Plug in the equation to Desmos, but split it up into something like y = 9x + 5 and y = a(x + b). Create sliders and substitute in the different statements, one by one. To test statement I, you are looking for what will create a parallel line to y = 9x + 5. Slide "a" around a bit, and you will see that as it approaches a value of 9, the two lines appear uniformly separated. You can zoom in to check if you want.

It must be true that the slope of a line that would produce no solutions would be 9, so statement I must be true. That one is not too bad. Next up, statement II. If a = 9 and b = 5, the two lines are still parallel, just a little farther apart.

While it is true that the above graph also shows no solution to the system we have created, that has nothing to do with b, but with the slope of the blue line. Comparing the two graphs, it is clear that b does not have to equal 5, since the default value of 1 also produced no solution to the system. At this point, you could safely eliminate answer choice (C), and only (B) and (D) would be in the running. Finally, set b to 5/9. I would recommend plugging in this value directly to avoid any confusion with decimals. Click on the default upper limit for b (10) and type in 5/9 instead.

The two lines have merged into one, meaning that the system would have an infinite number of solutions, rather than none, if a = 9 and b = 5/9. By test, then, you could have proved that b could be any value except for 5/9 to create a parallel line. This means that statement III must also be true, and the correct answer is (D).

Method 3: There is nothing wrong with working out the algebra. Just change the "equals" sign to a "does not equal" sign.

9x + 5 ≠ a(x + b)

9x + 5 ≠ ax + ab

For the x's to cancel out, a would have to be 9. Substitute:

9x + 5 ≠ (9)x + (9)b

5 ≠ 9b

5/9 ≠ b

Once again, we see that the correct answer is (D). The question takes a basic concept and presents a challenging spin by introducing extra variables, but good organization makes it look pretty simple.