As always, it’s important to think about the answer choices, and how they relate to the concepts and quantities in the question. Notice that every answer choice has a clear relationship to the quantities 7 and 5 from the question:
(A) is the difference of 7 and 5, so it would make sense as the right answer if x2y - y2x were the same thing as the difference between xy and x-y
(B) is the sum of 7 and 5, so it would make sense if x2y - y2x were the sum of xy and x-y.
(C) is twice the sum or 7 and 5, which also doesn’t make any sense here.
(D) is the product of 7 and 5, so it will be the right answer if x2y - y2x is the same thing as the product of xy and x-y. Which is exactly what it is, so (D) is correct.
(E) is twice the product of 7 and 5.
As I’ve mentioned many times, we really want to get in the habit of thinking of the answer choices as part of the question itself, not as an afterthought. Looking for ways that the choices relate back to the text of the question can really help you increase both your speed and your accuracy. This question is one of the most frequently missed questions in this section (we can tell this from the College Board’s difficulty ranking), but it can literally be answered without even picking up a pencil if we notice what’s going on in the answer choices. This question ultimately boils down to paying attention, knowing what 7 * 5 is, and knowing how to multiply simple expressions in algebra.
Page 953, Question 17
This is a grid-in question that asks us to find “one possible value of the slope” of a line that intersects line segment AB. Remember that when a question like this asks us to find “one possible value,” then there must be more than one value possible—whenever a question refers to the possibility of multiple values, there are multiple values.
In this case, any slope value of any line whatsoever that intersects AB will be fine. If a line (8,3) hits point A exactly, then a line through (9,3) must intersect AB. The line through (9,3) will have a slope of 3/9, or 1/3. So that’s one possible value.
Another way might be to try to cut AB in half. To do that, we could imagine a line through (8,1.5), or through (16,3) (that would be the same line, determined at two different, collinear, points). Either one would create a line with a slope of 3/16.
Yet another way to create a valid answer might be to realize that a line with a slope just barely over horizontal would have to intersect AB since point B lies on the x-axis. So we could pick a slope of 1/10 or .001 or something.
Any of these approaches could work, as could a wide variety of others.
Page 969, Question 16
I once saw a YouTube video of a guy who explained how to attack this question in the most complicated way I could possibly imagine. He began by identifying the curve in the graph as a parabola, and then tried to figure out what the equation for the parabola would be (after trying to figure out the coordinates of various points on the parabola, like the vertex, the intercepts, and so on). Once he had the equation of the parabola worked out, he set it equal to 0, solved it, and then found his answer.
It worked, of course, but it created a ridiculous amount of extra work for no reason—and let’s not even mention the extra opportunities for error it created by drawing the solution out unnecessarily.
Unfortunately, that kind of approach is what most test-takers would try on this question, because they’re programmed to answer questions on the SAT Math section the same way they would answer questions in school.
There’s a much easier way to approach this question, which relies on simply understanding the properties and definitions of concepts relating to the graphs of functions.
We should recognize that the places where h(a) = 0 will be the places where the graph of h(a) crosses the x-axis.
We should also recognize that the graph is drawn to scale, which means we can eyeball it relative to the provided values and figure out where the graph crosses the axis from there.
If we look at the answer choices, we’ll see that (A) must be correct, because where x = -1 on the given graph, we find h(a) crossing the x-axis.
Some students are tempted to choose 4 as the answer to this question, because it kind of looks like the right-hand x-intercept of the graph might be at x = 4. But if you look carefully, you’ll see it’s at a position much closer to x = 5. Remember how important it is to pay attention to details!
If we read carefully, look at the graph carefully, and remember to think in terms of things like intercepts, we can answer this question in about 10 seconds, without even picking up a pencil.
Page 981, Question 15
A lot of test-takers drop the ball on this question because they don’t look for the ways to simplify the algebraic expressions involved, or they try to assign values to n and k without making sure that those values satisfy the original requirements of the given equation.